# Schwinger, Tomonaga, Feynman, and Dyson: the triumph of renormalization

# Schwinger, Tomonaga, Feynman, and Dyson: the triumph of renormalization

# Abstract and Keywords

Barely six months after the Shelter Island Conference, which reawakened his interest in quantum electrodynamics (QED), and just three months after returning to Harvard University from his extended honeymoon to the West Coast, Julian Schwinger published a one-page note in the *Physical Review* entitled ‘On quantum electrodynamics and the magnetic moment of the electron’. A preliminary account of this work was presented by Schwinger at the 10th Washington Conference on Theoretical Physics in November 1947, which attracted the interest of J. Robert Oppenheimer and Richard Feynman. This chapter looks at Schwinger's method of canonical transformations, his covariant approach to QED, Sin-itiro Tomonaga's covariant formulation of quantum field theory, Feynman's theory of positrons and his space-time approach to quantum electrodynamics, Freeman Dyson's research on the radiation theories of Schwinger, Tomonaga, and Feynman, and the synergism between the works of Feynman and Schwinger with respect to QED.

*Keywords:*
quantum electrodynamics, Richard Feynman, Freeman Dyson, radiation theories, positrons, space-time, canonical transformations, Sin-itiro Tomonaga, quantum field theory

# Schwinger’s method of canonical transformations

Barely six months after the Shelter Island Conference, which reawakened his interest in quantum electrodynamics, and just three months after returning to Harvard from his extended honeymoon to the West Coast, Schwinger published a one-page note in the *Physical Review* entitled ‘On quantum electrodynamics and the magnetic moment of the electron’ [43]. A preliminary account of this work was presented by Schwinger at the 10th Washington Conference on Theoretical Physics in November 1947,* which, as we saw in the previous chapter, attracted the interest of Oppenheimer and Feynman.^{2} Schwinger recalled this meeting as the first time he actually significantly interacted with Feynman,^{1} while Feynman was impressed by Schwinger’s presentation on the anomalous magnetic moment of the electron, and on the Lamb shift. In the published
(p.252)
paper Schwinger stated the result of a calculation of ‘an additional magnetic moment associated with the electron spin, of magnitude’*

*α = e*

^{2}/

*ħe*≈ 1/137 the fine structure constant, a result completely consistent with the recent results on hyperfine splitting,

^{4}most precisely in agreement with Kusch and Foley’s results for sodium and gallium,

^{5}which could be interpreted as an additional magnetic moment for the electron of δμ/μ = 0.00118 ± 0.00003.

In the last of the four paragraphs of this one-page paper, he mentioned the result of a relativistic calculation of the Lamb shift. ‘The values yielded by our theory differ only slightly from those conjectured by Bethe^{6} on the basis of a nonrelativistic calculation,^{†} and are, thus, in good accord with experiment.’ As we know, all was not so well. ‘Finally, the finite radiative correction to the elastic scattering of electrons by the Coulomb field provides a satisfactory termination to a subject that has been beset with much confusion.’ This is a reference to the incorrect Dancoff calculation,^{7} to which we alluded in Chapter 3 and Chapter 7. ‘In 1939 Oppenheimer, I presume, suggested to Dancoff that he do a relativistic calculation of the electrodynamic corrections to scattering of an electron by a nucleus. He did that calculation and made a mistake, as a result of which it was not immediately obvious that all the electrodynamic corrections could be explained by uniting an electromagnetic mass with a mechanical mass. History might have been very different if that mistake had not been made. I think the Lamb shift could have been predicted.’^{1}

Schwinger concluded by promising a paper detailing the theory and the applications. Alas, that was not to come to pass. By the time of the Pocono Conference (p.253) four months later, he had already constructed a covariant formulation, making the technique underlying this first paper obsolete.

It is important to recognize that, of course, Schwinger was well aware of the problems of electrodynamics from his earliest student days.* Moreover, he wrote a paper when he was 16, which he never submitted to a journal, entitled ‘On the interaction of several electrons’ [0], which discussed the M0ller interaction,^{9} based on the Dirac–Fock–Podolsky electrodynamics of 1932;^{10} Schwinger’s first effort was noteworthy for the introduction of the interaction representation [199]. Later, when he went to Berkeley, he discovered that Oppenheimer was obsessed with the subject. Indeed, Oppenheimer and Schwinger wrote a joint paper on ‘Pair emission in the proton bombardment of fluorine’ [15], where the explanation of the observed effect turned out to be the existence of vacuum polarization, the virtual creation, for short periods of time, of electron-positron pairs. (Although, to Schwinger’s annoyance, Oppenheimer insisted on inserting remarks about a possible (non-existent) non-electromagnetic coupling between electrons and nuclear particles.) Thus he began with an advantage over Feynman, who failed to recognize the reality of vacuum polarization for the first few years of the development of quantum electrodynamics.

Crucial for Schwinger’s stunning progress in quantum electrodynamics after the war was his development of electromagnetic theory at the MIT Radiation Laboratory, and in particular his perfection of the theory of synchrotron radiation immediately after the end of the war. ‘What was significant was the radiation emitted by relativistic electrons moving in circular paths under magnetic field guidance. It is an old problem, but the quantitative implication of relativistic energies had not been appreciated. In attacking this classical relativistic situation, I used the invariant proper-time formulation of action, including the electromagnetic self-action of a charge. That self-action contained a resistive part and a reactive part, to use the engineering language I had learned. The reactive part was the electromagnetic mass effect, here automatically providing an invariant supplement to the mechanical action and thereby introducing the physical mass of the charge. Incidentally, in the paper on synchrotron radiation that was published several years later, a more elementary expression of this method is used, and the reactive effect is dismissed as “an inertial effect with which we are not concerned” [56,37]. But here was my reminder that electromagnetic self-action, physically necessary in one context, was not to be, and need not be, omitted in another context. And in arriving at a relativistically invariant result, in a subject where relativistic invariance was notoriously (p.254) difficult to maintain, I had learned a simple but useful lesson: to emerge with relativistically invariant physical conclusions, use a covariantly formulated theory, and maintain covariance throughout the calculation/ [199]

Hendrik Kramers is usually mentioned as the father of the concept of renor-malization. Yet his approach was overly classical.* ‘Of course, the concept of electromagnetic self-action, of electromagnetic mass, had not entirely died out in that age of subtraction physics; it had gone underground, to surface occasionally. Hans Kramers must be mentioned in this connection. In a book published in 1938 he suggested that the correspondence-principle foundation of quantum electrodynamics was unsatisfactory because it was not related to a classical theory that already included the electromagnetic mass and referred to the physical electron.^{11} He proposed to produce such a classical theory by eliminating the proper field of the electron, the field associated with uniform motion. Very good—if we lived in a nonrelativistic world. But it was already known from the work of Victor Weisskopf and Wendell Furry that the electromagnetic mass problem is entirely transformed in the relativistic theory of electrons and positrons, then described in the unsymmetrical hole formulation—the relativistic electromagnetic mass problem is beyond the reach of the correspondence principle.^{12}^{†} Nevertheless, I must give Kramers very high marks for his recognition that the theory should have a structure-independent character. The relativistic counterpart of that was to be my guiding principle, and over the years it has become generalized to this commandment: Thou shalt not entangle that which is known, and reliable, with that which is unknown, and speculative. The effective range treatment of nuclear forces, which evolved just after the war, also abides by this philosophy [40, 58]’ [199]. In a book review, Schwinger summarized his position on Kramers: ‘It is a common mistake to think that Kramers had anticipated post-war mass renormalization. His idea was to begin with the classical nonrelativistic Hamiltonian expansion in terms
(p.255)
of the physical mass, and quantize it. But quantum relativistic effects change the nature of this self-mass.’^{8}

We will not discuss Schwinger’s first approach to quantum electrodynamics in any detail, in part because it was so quickly superseded by covariant methods, and thus of only historical importance, and secondly because it is discussed in mathematical detail in Schweber’s book.^{2} (Notes on this development may be found in the UCLA archive.^{8}) It consists of the application of canonical or contact transformations to isolate the effects of mass and charge renormalization.

As a result of this remarkable advance, Schwinger was invited to give a lecture at the 1947 Annual Meeting of the American Physical Society, which took place at Columbia University at the end of January 1948. Schwinger’s lecture took place on Saturday 31 January. That meeting, and Feynman’s interaction with Schwinger there, was described in Chapter 7. No printed or manuscript version of the lecture apparently exists, but it was by all accounts a brilliant success.* In the minutes of the meeting, Karl K. Darrow, secretary of the APS, noted the unprecedented event that Schwinger’s lecture ‘was repeated by popular request.’^{13} In his diary, Darrow stated that he ‘heard no paper but Schwinger’s, given too rapidly for my apprehension but given with great gusto which implied a great advance.’^{14} Freeman Dyson was in attendance, and in writing about it to his parents noted the birth of a new formulation: ‘The great event came on Saturday morning, and was an hour’s talk by Schwinger, in which he gave a masterly survey of the new theory which he has the greatest share in constructing and at the end made a dramatic announcement of a still newer and more powerful theory, which is still in embryo. This talk was so brilliant that he was asked to repeat it in the afternoon session, various unfortunate lesser lights being displaced in his favour. There were tremendous cheers when he announced that the crucial experiment had supported his theory; the magnetic splitting of two of the spectral lines of gallium … were found to be in the ratio 2.00114 [sic] to 1; the old theory gave for this ratio exactly 2 to 1, while the Schwinger theory gave 2.0016 [sic] to l.’^{15}

Indeed, by the time of the January meeting, the new covariant theory was far advanced. Again, in Schwinger’s words, ‘The third stage, the development of the first covariant theory, had already begun at the time of the New York meeting in
(p.256)
January. I have mentioned that the simple idea of the interaction representation had presented itself 14 years earlier, and the space-time treatment of both electromagnetic and electron-positron fields was inevitable. I have a distinct memory of sitting on the porch of my new residence [in his wife’s mother’s house] during what must have been a very late Indian summer in the fall of 1947 and with great ease and great delight arriving at invariant results in the electromagnetic-mass calculation for a free electron. I suspect this was done with an equal-time interaction.* The spacelike generalization, to a plane, and then to a curved surface, took time, but all that was in place at the New York meeting. I must have made a brief reference to these covariant methods; the typed copy [of Ref. [43]] contains such an equation on another back page, and I know that Oppenheimer told me about Sin-itiro Tomonaga after my lecture.^{†} [199]

On a human note, we also recall from Chapter 2 the story of Rabi’s teasing of the unfortunate Professor LaMer in the elevator in the Faculty Club at Columbia, after the third repeated lecture; LaMer was the only man who had dared to flunk the prodigy Schwinger for not following the rules.

# Schwinger’s covariant approach

The covariant approach to quantum electrodynamics, which Schwinger presented in “Quantum electrodynamics. I” [50], “II” [52], and “III” [57] was essentially identical to that first described at the Pocono Conference, at the Washington Meeting of the American Physical Society [47], also held in April, 1948, and then given in detail at the Michigan Summer School that year. These presentations have been recounted in the previous chapter. These papers were also the basis for his successful application for the Charles L. Mayer Nature of Light Award in October of that year, which we described in the previous chapter. The first of these papers was submitted just over six months after his announcement of the solution of the problems of quantum electrodynamics in [43], in July of 1948, with the second and third reaching the hands of the editors of *Physical Review* in November, and the following May, respectively.

Why was it necessary for Schwinger to abandon the non-covariant approach which so successfully yielded the *a/2n* correction to the magnetic moment of
(p.257)
the electron? It was the difficulty of correctly carrying out a relativistic calculation of the Lamb shift, that is, the electrodynamic displacement of hydrogen energy levels from the values predicted by the Dirac equation. Although Schwinger advertized in his note [43] success on this front, it was not satisfactory. Let us quote Schwinger himself, from his introductory remarks in his collection of the most important papers in the field, *Quantum electrodynamics* [83]: first, he recounted the progress since Kramers,^{11} spurred by experiment. ‘Exploiting the wartime development of electronic and microwave techniques, delicate measurements disclosed that the electron possessed an intrinsic magnetic moment slightly greater than that predicted by the relativistic quantum theory of a single particle,^{5} while another prediction of the latter theory concerning the degeneracy of states in the excited levels of hydrogen was contradicted by observing a separation of the states.^{16} (Historically, the experimental stimulus came entirely from the latter measurement; the evidence on magnetic anomalies received its proper interpretation only in consequence of the theoretical prediction of an additional spin magnetic moment [by Schwinger].) If these new electron properties were to be understood as electrodynamic effects, the theory had to be recast in a usable form. The parameters of mass and charge associated with the electron in the formalism of electrodynamics are not the quantities measured under ordinary conditions. A free electron is accompanied by an electromagnetic field which effectively alters the inertia of the system, and an electromagnetic field is accompanied by a current of electron-positron pairs which effectively alters the strength of the field and of all charges. Hence a process of renormalization must be carried out, in which the initial parameters are eliminated in favor of those with immediate physical significance. The simplest approximate method of accomplishing this is to compute the electrodynamic corrections to some property and then subtract the effect of the mass and charge redefinitions. While this is a possible nonrelativistic procedure,^{6} it is not a satisfactory basis for relativistic calculations where the difference of two individually divergent terms is generally ambiguous. It was necessary to subject the conventional Hamiltonian electrodynamics to a transformation designed to introduce the proper description of single electron and photon states, so that the interaction among these particles would be characterized from the beginning by experimental parameters. As a result of this calculation [43], performed to the first significant order of approximation in the electromagnetic coupling, the electron acquired new electrodynamic properties, which were completely finite. These included an energy displacement in an external magnetic field corresponding to an additional spin magnetic moment, and a displacement of energy levels in a Coulomb field. Both predictions were in good accord with experiment, and later refinements in experiment and theory have only emphasized that agreement.’

(p.258)
But the calculation of the energy shift in the field of the nucleus, the Coulomb field, revealed the deficiency in the technique. Schwinger went on, ‘However, the Coulomb calculation disclosed a serious flaw; the additional spin interaction that appeared in an electrostatic field was not that expected from the relativistic transformation properties of the supplementary spin magnetic moment, and had to be artificially corrected.^{17}’^{18}’* Thus, a complete revision in the computational techniques of the relativistic theory could not be avoided. The electrodynamic formalism is invariant under Lorentz transformations and gauge transformations, and the concept of renormalization is in accord with these requirements. Yet, in virtue of the divergences inherent in the theory, the use of a particular coordinate system or gauge in the course of computation could result in a loss of covariance. A version of the theory was needed that manifested covariance at every stage of the calculation. The basis of such a formulation was found in the distinction between the elementary properties of the individual uncoupled fields and the effects produced by the interaction between them^{19} [50]. The application of these methods to the problems of vacuum polarization, electron mass, and the electromagnetic properties of single electrons now gave finite, covariant results which justified and extended the earlier calculations [57]. Thus, to the first approximation at least, the use of a covariant renormalization technique had produced a theory that was devoid of divergences and in agreement with experience, all high energy difficulties being isolated in the renormalization constants. Yet, in one aspect of these calculations, the preservation of gauge invariance, the utmost caution was required,^{20} and the need was felt for less delicate methods of evaluation. Extreme care would not
(p.259)
be necessary if, by some device, the various divergent integrals could be rendered convergent while maintaining their general covariant features. This can be accomplished by substituting, for the mass of the particle, a suitably weighted spectrum of masses, where all auxiliary masses eventually tend to infinity.^{21} Such a procedure has no meaning in terms of physically realizable particles. It is best understood, and replaced, by a description of the electron with the aid of an invariant proper-time parameter. Divergences appear only when one integrates over this parameter, and gauge invariant, Lorentz invariant results are automatically guaranteed merely by reserving this integration to the end of the calculation [64].’ This last remark was a reference to the well-known Pauli-Villars regularization technique, and Schwinger’s reaction to it, the magnificent ‘Gauge invariance and vacuum polarization’ [64] paper, which we shall describe in detail in the following chapter.

However, at first Schwinger’s covariant calculation of the Lamb shift contained another error, the same as Feynman’s.^{22} ‘By this time I had forgotten the number I had gotten by just artificially changing the wrong spin-orbit coupling. Because I was now thoroughly involved with the covariant calculation and it was the covariant calculation that betrayed me, because something went wrong there as well. That was a human error of stupidity.’ French and Weisskopf^{23} had gotten the right answer, ‘because they put in the correct value of the magnetic moment and used it all the way through. I, at an earlier stage, had done that, in effect, and also got the same answer.’ But now he and Feynman ‘fell into the same trap. We were connecting a relativistic calculation of high energy effects with a nonrelativistic calculation of low energy effects, *a la* Bethe.’ Based on the result Schwinger had presented at the APS meeting in January 1948, Schwinger claimed priority for the Lamb shift calculation: ‘I had the answer in December of 1947. If you look at those [other] papers you will find that on the critical issue of the spin-orbit coupling, they appeal to the magnetic moment. The deficiency in the calculation I did [in 1947] was [that it was] a non-covariant calculation. French and Weisskopf were certainly doing a non-covariant calculation. Willis Lamb^{24} was doing a non-covariant calculation. They could not possibly have avoided these same problems.’^{1} The error Feynman and Schwinger made had to do with the infrared problem that occurred in the relativistic calculation, which was handled by giving the photon a fictitious mass. ‘Nobody thought that if you give the photon a finite mass it will also affect the low energy problem. There are no longer the two transverse degrees of freedom of a massless photon, there’s also a longitudinal degree of freedom. I suddenly realized this absolutely stupid error, that a photon of finite mass is a spin-1 particle, not a helicity-1 particle.’^{1}

An indication of the impact of Schwinger’s breakthroughs, as seen by his peers at the time, is given by J. Robert Oppenheimer’s remarks at the 1948 Solvay Conference in Brussels, to which Schwinger was invited, but did not attend, due to
(p.260)
some mixup in the invitation. After reviewing the failures of the old quantum field theory, Oppenheimer stated, ‘Such a procedure would no doubt be satisfactory, if cumbersome, were all quantities involved finite and unambiguous. In fact, since mass and charge corrections are in general represented by logarithmically divergent integrals, the above outlined procedure serves to obtain finite, but not necessarily unique or correct, reactive corrections for the behavior of an electron in an external field; and a special tact is necessary, such as that implicit in Luttinger’s derivation^{25} of the electron’s anomalous gyromagnetic ratio, if results are to be, not merely plausible, but unambiguous and sound. Since, in more complex problems, and in calculations carried to higher orders in e, this straightforward procedure becomes more and more ambiguous, and the results are more dependent on the choice of Lorentz frame and of gauge, more powerful methods are required. Their development has occurred in two steps, the first largely, the second wholly, due to Schwinger [50].’^{18}

‘Quantum electrodynamics. I. A covariant formulation’ [50] received by *Physical Review* on 29 July 1948, is a comprehensive development of the theory first presented in an all-day* talk at the Pocono Conference on 31 March of that year, which we described in the in preceding chapter, and which, with the first draft of ‘Quantum electrodynamics IF [52] was used as the basis for his lectures at the Michigan Summer School during the period 19 July through 7 August 1948. Among those present in the audience for the Michigan lectures was again Freeman Dyson.

The paper begins with an extended abstract that summarizes the matter brilliantly: Attempts to avoid the divergence difficulties of quantum electrodynamics by multilation of the theory have been uniformly unsuccessful. The lack of convergence does indicate that a revision of electrodynamic concepts at ultrarel-ativistic energies is indeed necessary, but no appreciable alteration of the theory for moderate relativistic energies can be tolerated. The elementary phenomena in which divergences occur, in consequence of virtual transitions involving particles with unlimited energy, are the polarization of the vacuum and the self-energy of the electron, effects which essentially express the interaction of the electromagnetic and matter fields with their own vacuum fluctuations. The basic result of these fluctuation interactions is to alter the constants characterizing the properties of the individual fields, and their mutual coupling, albeit by infinite factors. The question is naturally posed whether all divergences can be isolated in such unobservable renormalization factors; more specifically, we inquire whether quantum electrodynamics can account unambiguously for the recently observed deviations from the Dirac electron theory, without the introduction of fundamentally new concepts. This paper, the first in a series devoted
(p.261)
to the above question, is occupied with the formulation of a completely covariant electrodynamics. Manifest covariance with respect to Lorentz and gauge transformations is essential in a divergent theory since the use of a particular reference system or gauge in the course of calculation can result in a loss of covariance in view of the ambiguities that may be the concomitant of infinities. It is remarked, in the first section, that the customary canonical commutation relations, which fail to exhibit the desired covariance since they refer to field variables at equal times and different points of space, can be put in covariant form by replacing the four-dimensional surface *t* = const. by a space-like surface. The latter is such that light signals cannot be propagated between any two points on the surface. In this manner, a formulation of quantum electrodynamics is constructed in the Heisenberg representation, which is obviously covariant in all its aspects. It is not entirely suitable, however, as a practical means of treating electrodynamic questions, since commutators of field quantities at points separated by a time-like interval can be constructed only by solving the equations of motion. This situation is to be contrasted with that of the Schr?odinger representation, in which all operators refer to the same time, thus providing a distinct separation between kinematical and dynamical aspects. A formulation that retains the evident covariance of the Heisenberg representation, and yet offers something akin to the advantage of the Schr?dinger representation, can be based on the distinction between the properties on non-interacting fields, and the effects of coupling between fields. In the second section, we construct a canonical transformation that changes the field equations in the Heisenberg representation into those of non-interacting fields, and therefore describes the coupling between fields in terms of a varying state vector. It is then a simple matter to evaluate commutators of field quantities at arbitrary space-time points. One thus obtains an obviously covariant and practical form of quantum electrodynamics, expressed in a mixed Heisenberg-Schr?dinger representation, which is called the interaction representation. The third section is devoted to a discussion of the covariant elimination of the longitudinal field, in which the customary distinction between longitudinal and transverse fields is replaced by a suitable covariant definition. The fourth section is concerned with the description of collision processes in terms of an invariant collision operator, which is the unitary operator that determines the overall change in state of a system as a result of interaction. It is shown that the collision operator is simply related to the Hermitian reaction operator, for which a variational principle is constructed.’

The interaction representation indeed seems to have been Schwinger’s invention, although he notes in a footnote that ‘The interaction representation can be regarded as a field generalization of the many-time formalism, from which point of view it has already been considered by S. Tomonaga.’^{19} In that representation,
(p.262)
the evolution of the state vector Ψ on a particular spacelike surface σ is given by a covariant Schr?dinger equation,

*H*is the interaction Hamiltonian,

*j*

_{μ}being the electric current density of the electrons, and

*A*

_{μ}the electromagnetic vector potential. (In this paper, and implicitly in all his early papers, Schwinger used an imaginary fourth-component of the four-vector position: χ

_{μ}= (r,

*ict*).) This evolution equation would later be referred to by Oppenheimer as the Tomonaga equation. Indeed, as we will see, exactly this equation, in very similar notation, appears in Tomonaga’s 1946 paper,

^{19}to which Schwinger refers.

Schwinger’s first paper is largely devoted to setting up the machinery. Most interesting, perhaps, is the final section, which begins with the words: ‘While the interactions between fields and their vacuum fluctuations are conveniently regarded as modifying the properties of the non-interacting fields, other types of interactions are often best viewed as producing transitions among the states of the individual fields. We shall conclude this paper with a brief discussion of a covariant manner of describing such transitions.’ Thus, the state vector on an arbitrary spacelike surface σ is related to that on an initial surface σ_{1} by a unitary operator:

*U*satisfies the equation of motion,

This differential equation is equivalent to a functional integral equation,

_{1}and σ. If we let those surfaces recede to ∓=∞, respectively, we obtain (p.263) the collision operator 5, which ‘determines the overall change in state of the system as the result of interaction,’

This unitary operator may be written in terms of a Hermitian reaction operator *K*,

Schwinger concludes this paper by showing that *K* satisfies a variational principle, of a type which he had used in scattering problems [40,49], which we described in Chapter 5—see Eqn. (5.31).

‘Quantum electrodynamics. II. Vacuum polarization and self-energy’ [52] reached the editors of *Physical Review* on 1 November 1948. Now Schwinger got down to work: ‘The covariant formulation of quantum electrodynamics, developed in a previous paper, is here applied to two elementary problems—the polarization of the vacuum and the self-energies of the electron and photon.’ He first defined ‘the vacuum of the isolated electromagnetic field to be that state for which the eigenvalue of the energy, or better, an arbitrary time-like component of the energy-momentum four-vector, is an absolute minimum.’ In that state, the energy-momentum tensor has vanishing expectation value, ‘the only result compatible with the requirement that the properties of the vacuum be independent of the coordinate system,’ because the energy-momentum tensor the electromagnetic field is traceless. As for the matter—that is, the electron—fields, the vacuum must be such that the vacuum expectation value of the electromagnetic current density vanish; while the vacuum expectation value of the electron energy-momentum tensor is not necessarily zero, but can be so redefined.

Armed with these properties, Schwinger went on to compute the polarization of the vacuum. That is, as a consequence of fluctuations in the electron-positron fields, the vacuum expectation value of the electromagnetic current is no longer zero in the presence of an external current *J*μ. The result is particularly simple if the latter is time independent, and then has the form

^{r}is an element of volume, and (unlike Schwinger, we set

*ħ = c*= 1)

Note that Schwinger in 1948 is using what would later universally be referred to as a Feynman parameter *v*, which Feynman would introduce only in 1949 to
(p.264)
combine his propagators in momentum space.^{26}’* This result can be expressed as a correction to the Coulomb potential, for short distances,

^{27}(See Eqn (6.25).)

Schwinger next went on to calculate the self-energy of the electron. The outcome was a ‘logarithmically divergent result for the electromagnetic mass of the electron or positron.’ Either by using the lower limit of a parameter integral, *w* _{0} → 0, or a large momentum scale, *K* → ∞, to define the divergent integral, he found for the ratio of the electromagnetic mass *δ m* to the bare mass *m* _{0}

*m*=

*m*

_{0}+

*δm*may be consistently used as the actual electron mass. (As we will see, this result, which was first derived in the hole theory by Weisskopf

^{12}with Furry’s help

^{2}(See Eqn (6.23)), was actually given a covariant derivation nearly six months earlier by Feynman.

^{22})

The old guard in Europe was not altogether satisfied with Schwinger’s breakthroughs. Gregor Wentzel objected to Schwinger’s claim at the Pocono Conference that the photon self-energy vanished; in the meantime Schwinger had developed an improved treatment of this question, which he had presented at the Michigan Summer School, and which appears in QED II, but Wentzel still had mathematical objections.^{28}

Not surprisingly, more confrontational was the reaction of Wolfgang Pauli. Schwinger sent a copy of QED II to him, and Pauli wrote back a detailed letter in January 1949. Pauli also objected to certain details of the vacuum polarization calculation, and strongly advocated his own regularization technique,^{21} which, as we have seen, Schwinger loathed. An extract of this letter appears in Schweber’s book.^{2} Schwinger did not reply, but rather passed the letter on to his student Bryce DeWitt, who responded without consulting Schwinger further. It was a reasonable argument involving the requirement of gauge invariance. Pauli then wrote a caustic letter to Oppenheimer: ‘My discussion with Schwinger, in
(p.265)
which he never participated himself, makes me think on “His Majesty’s” psychology. (An evening seminar on this subject—ladies admitted—would be very funny. I can also tell experimental material from earlier times.) His Majesty permitted one of his pupils (B. Seligmann) [B. DeWitt] to break the “blockade” of the ETH/Zurich by Harvard and write to me a letter, but he refused to read the letter himself! [In fact, DeWitt never showed Schwinger the letter.] The content of this diplomatic note (it was a very long one) is only this, that His Majesty had a kind of revelation on some Mt. Sinai, to put always, ∂ Δ^{(1)}/∂*x* _{v} = 0 for *x* = 0 (in contrast to ∂δ(*x*)/∂*x* _{v} which has same symmetry properties) wherever it occurs. We are calling here this equation “the revelation” but it did not help our understanding. The B. Seligmann and also a Mr. Glauber want to come here next spring, but both are unable to obtain a scientific recommendation from His Majesty who prefers to “sacrifice” both of them rather than write to me. I am enjoying this situation very much.’^{29} In fact, Schwinger did write strong letters of recommendation for both DeWitt and Glauber, and the following summer Schwinger visited Pauli in Zurich in an attempt to smooth ruffled feathers. We will describe that visit in the next chapter.

Six months after writing ‘QED II’, Schwinger submitted the third of this monumental series, ‘Quantum electrodynamics. III. The electromagnetic properties of the electron—radiative corrections to scattering’ [57]. It is important to recognize that Schwinger was also involved in several other completely independent projects at the same time. He submitted a paper on diffraction [54] with Harold Levine in January 1949 (and a correction [55] to an earlier paper with Levine in March), and submitted the important ‘Classical radiation of accelerated electrons’ [56] to *Physical Review* in March as well. (We have discussed these papers in Chapter 4 and chapter 5.) But clearly QED was now the focus. It may be helpful to quote the opening paragraphs of ‘QED III’: ‘A covariant form of quantum electrodynamics has been developed, and applied to two elementary vacuum fluctuation phenomena in the previous articles of this series. These applications were the polarization of the vacuum, expressing the modifications in the properties of the electromagnetic field arising from its interaction with the matter field vacuum fluctuations, and the electromagnetic mass of the electron, embodying the corrections to the mechanical properties of the matter field, in its single particle aspect, that are produced by the vacuum fluctuation of the electromagnetic field. In these problems, the divergences that mar the theory are found to be concealed in unobservable charge and mass renormalization factors.

‘The previous discussion of the polarization of the vacuum was concerned with a given current distribution, one that is not affected by the dynamical reactions of the electron-positron field. We shall now consider the more complicated situation in which the original current is that ascribed to an electron (p.266) or positron—a dynamical system, and an entity indistinguishable from the particles associated with the matter field vacuum fluctuations. The changed electromagnetic properties of the particle will be exhibited in an external field, and may be compared with the experimental indications of deviations from the Dirac theory that were briefly discussed in [QED] I. To avoid a work of excessive length, this discussion will be given in two papers. In this paper we shall construct the current operator as modified, to the second order, by the coupling with the vacuum electromagnetic field. This will be applied to compute the radiative correction to the scattering of an electron by a Coulomb field [53]. The second paper will deal with the effects of radiative corrections on energy levels.’ However, that second paper was never written, largely because soon Schwinger would begin work on a third reformulation of quantum electrodynamics, which we shall describe in the following chapter.

Let us concentrate on the results given in this monumental paper. After removing a spurious infrared divergence, Schwinger obtained first the additional spin magnetic moment he had first given a year and a half earlier,

*Ze*(a nucleus) through an angle ϑ, in which the energy loss (due to unseen low-energy photons radiated) is less than an amount Δ

*E*,

*βc*its speed. A general expression for the radiative correction δ is given; for a slowly moving particle it takes on the simple form

*E*is the electron energy. Schwinger concluded that these radiative corrections could amount to several per cent at the energies then available.

The above result (8.16) had, in fact, been derived nearly six months earlier. In January 1949, Schwinger had written a Letter to the *Physical Review* in which he discussed ‘Radiative corrections to electron scattering’ [53] There, he also discussed the Lamb shift, mentioned his earlier error due to the improper magnetic moment contribution, and stated that the result, equal to 1051 MHz for
(p.267)
the splitting of the 2^{2}S_{1/2} and 2^{2}P_{1/2} levels of hydrogen, was now in agreement with the calculations of French and Weisskopf^{23} and of Kroll and Lamb.^{24} As we remarked above, when Schwinger first did the covariant calculation, he made an additional error in matching the high- and low-energy contributions. Feynman made the same mistake,^{22} resulting in a significant delay in publication of the French and Weisskopf paper. Schweber criticized Schwinger for being less than forthright in acknowledging his error, unlike Feynman;^{2} but in Schwinger’s defense it should be noted that he never published his wrong result, giving the incorrect formula only at the Michigan Summer School.^{8} Schwinger gave a detailed account of his and Feynman’s ‘goof’ in his historical talk ‘Renor-malization theory of quantum electrodynamics: an individual view’ [199] in which he concluded, ‘And so, although Weisskopf was not the first to find the correct result, he was the first to insist on its correctness.’

We will have more to say about the story of the Lamb shift when we turn to Feynman.

# Tomonaga’s covariant formulation of quantum field theory

Amid the devastation of the last years of the war in Japan, and in the immediate post-war period, a remarkable group around Sin-itiro Tomonaga in Tokyo made enormous progress in formulating a consistent, Lorentz invariant quantum electrodynamics, although the participants were completely cut off from the developments in America. This approach turned out to be remarkably similar to the covariant approach of Schwinger, although, in large measure due to their isolation, they were not able to carry the program fully through to a theory capable of producing reliable calculations. The West learned of this work through Tomonaga’s communication to Oppenheimer, delivered by hand by the first Japanese students to visit the US after the war, particularly Katsumi Tanaka.* Oppenheimer arranged for a brief note to be published in the *Physical Review by* Tomonaga.^{20} Oppenheimer evidently thought highly of Tomonaga’s work, and referred to it in glowing terms in his Solvay report.^{18} He attributed the origin of what Schwinger referred to as the ‘interaction representation’ to Tomonaga, and wrote down what he called the Tomonaga equation, namely Eqn. (8.2). He then described Schwinger’s program as removing the Virtual’ transitions from the right-hand side of this equation by contact transformations.

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Dyson recalled that in Spring 1948 Tomonaga sent two copies of *Progress of Theoretical Physics* to Bethe. The second issue included Tomonaga’s paper,^{19} which contained a remarkable footnote stating that the work had been published in Japanese in 1943.^{33} Dyson went on to say, ‘The implications of this were astonishing. Somehow or other, amid the ruin and turmoil of the war, totally isolated from the rest of the world, Tomonaga had maintained in Japan a school of research in theoretical physics that was in some respects ahead of anything existing anywhere else at that time. He had pushed on alone and laid the foundations of the new quantum electrodynamics, five years before Schwinger and without any help from the Columbia experiments. He had not, in 1943, completed the theory and developed it as a practical tool. To Schwinger rightly belongs the credit for making the theory into a coherent mathematical structure. But Tomonaga had taken the first essential step. There he was, in the spring of 1948, sitting amid the ashes and rubble of Tokyo and sending us that pathetic little package. It came to us as a voice out of the deep.’^{34}

What was the background of this remarkable accomplishment? We cannot, in a short space, do justice to the achievements of Sin-itiro Tomonaga. A brief history of his life and accomplishments is given in Schweber’s book,^{2} and further details may be found in the collected memoirs and reminiscences edited by Makinosuke Matsui.^{35} The following synopsis of Tomonaga’s life is extracted from Matsui’s contributions to that book.

Tomonaga was born in Tokyo in 1906, the first son of a philosophy professor, Sanjuro Tomonaga, and his wife Hide. The next year his father was offered a post at Kyoto Imperial University, so the family moved to Kyoto. In 1909 Sanjuro went to Heidelberg to study, and remained there for four years; the family stayed in Tokyo with Hide’s parents until 1913. On the father’s return that year, they moved back to Kyoto, where they lived on the grounds of the Shogoin temple for more than a decade. Tomonaga was a sensitive child, in his own words, a ‘crybaby,’ and suffered from poor health. As a result, the family had to spend expensive summer vacations at seaside resorts. In 1918 Tomonaga graduated from Kinrin Elementary School, and enrolled in Kyoto First Middle School, a premiere academically-oriented school. Because his family was of samurai lineage, he was brought up strictly, in spite of his delicate health. The affection he did not receive from his immediate family, he received from his uncle, Masuzo Tomonaga.*

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Tomonaga recalled that in elementary school he was not very good at physical education. ‘But once, after we had run a lap around Heian Shrine in a sort of marathon, my teacher praised me because I had managed to run the entire course without dropping out. Running the course through was nothing extraordinary and dropping out was actually unusual, mind you, but children are always delighted to be praised whether or not what they have done is remarkable.’^{36} He had fond memories of chemistry demonstrations in school. He initially had some difficulty in middle school, largely because he missed the entire first term because of illness. He recalled being stimulated by his mathematics teacher’s educational style there. He also remembered a science teacher telling them about radioactivity, and the uranium that supposedly could be found in the mountains around Kyoto. A group of students decided to go on an expedition to find some, but Tomonaga caught a cold and was unable to go; in any case no uranium was found, at least no sample that glowed in the dark.

His brother, Yojiro Tomonaga, recalled that Tomonaga liked to make models and craft objects, even as an adult, which he attributed to his reading the magazine *Science for boys*. He also liked to take trick photographs. He started to paint from nature as a child.^{37} Schweber recounts Tomonaga’s early electric and optical projects, in which he built everything from scratch.^{2}

In 1923 Tomonaga entered the Third High School, in Science Department B.* In high school, he still missed classes frequently because of illness. Although quiet, he was not always well-behaved; Masatada Tada recalled that he once smeared chalk dust all over a teacher’s chair, but confessed and accepted the punishment willingly.^{35}

From high school, Tomonaga was admitted in 1926 to the Faculty of Science of the Kyoto Imperial University. Fellow students included Hideki Yukawa, who had attended the same middle school as Tomonaga and who would later achieve fame for proposing the existence of the meson. Yukawa and Tomonaga learned quantum mechanics on their own by reading the original papers. After graduation from the University in 1929, Tomonaga and Yukawa stayed on as unpaid assistants. Tomonaga recalled attending lectures of Dirac and Heisenberg in Tokyo.^{2} But in 1930 Yoshio Nishina returned from Europe, where he had been studying with Niels Bohr, and gave a lecture in Kyoto based on Heisenberg’s book.^{38} Tomonaga was inspired, and asked penetrating questions of Nishina,
(p.270)
who as a result offered him a position at the Institute of Physical and Chemical Research (Riken) in Tokyo the following year (1932). After a three-month trial period, Tomonaga, with some agonizing, joined the institute on a permanent basis, and gradually came out of his shell, and engaged in sports and social life.

There was an exchange agreement between the institute and Leipzig University, and in 1937 Tomonaga went to Leipzig to study with Heisenberg. Apparently, he was rather depressed there [200]. Two years later, he returned and, the following year, married Ryoko Sekiguchi. In 1939 he became a professor at Tokyo Bunrika University. There, he started the research program that eventually earned him the Nobel Prize. As the war intensified, the Japanese navy asked him to work on radar, and he developed a powerful magnetron, starting in 1943. In one of the remarkable parallels of history, at the same time, Julian Schwinger, in Cambridge, Massachusetts, was carrying out very similar work on microwave cavities. Both men independently developed the theory of the *S*, or scattering, matrix for waveguides, which would later have important implications for field theory and particle physics. Of course, at the time, neither had any knowledge that the other existed.

Just after the war, with the US Army occupying Japan, an American soldier drove up to Tokyo Bunrika University, asking for Professor Tomonaga. Since arrests of war criminals were in the news, some alarm was registered. It turned out that the soldier was physicist Philip Morrison, who had been involved in the dropping of the first bomb on Hiroshima, and was visiting to assess the damage wrought by the nuclear explosion. He was merely calling on Tomonaga to express his regards.

In 1946 Tomonaga won the Asahi Prize, for his work on meson theory and on the super-many-time theory, the proceeds of which (10,000 yen) he used to buy tatami mats to furnish a miserable abandoned building on the Okubo campus (which had previously belonged to the Japanese Imperial Army) for his family’s residence. His group was already established in another concrete building on this site. For a graphic account of the facilties, see the article by Daisuki Ito in Ref. 35.

# Tomonaga’s papers

Tomonaga’s 1946 paper in *Progress of Theoretical Physics*^{19} was entitled ‘On a relativistically invariant formulation of the quantum theory of wave fields,’ and was noted as having first been published in Japanese in 1943.^{33} It generalized the Schr?dinger equation by proceeding from the many-time formulation of Dirac.^{39} That is, there were as many time variables as there were particle coordinates in the state vector. This suggested the introduction of infinitely many time variables, one for each space point, *t* _{xyz}, a local time, an idea which had also been introduced by Stückelberg.^{40} From this perspective, he was able to
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define the state vector as a functional of the space-like surface *C*,Ψ(*C*), which satisfied the functional Schr?dinger equation

Here *H* _{12} is the interaction Hamiltonian between the two fields that Tomonaga was considering and *Cp* is a surface passing through the point *P*. Indeed, this is the same equation (8.2) that Schwinger would obtain five years later, in 1948, in his ‘Quantum electrodynamics I’ [50]. A form equivalent to the integral equation (8.7) was also given by Tomonaga in this paper, with nearly the same notation. Tomonaga rounded out the paper by giving generalized probability amplitudes and transformation functions.

To assess this work in context, it may be useful to quote the ‘Concluding remarks’: ‘We have thus shown that the quantum theory of wave fields can be really brought into a form which reveals directly the invariance of the theory against Lorentz transformations. The reason why the ordinary formalism of the quantum field theory is so unsatisfactory is that it has been built up in a way much too analogous to the ordinary nonrelativistic mechanics. In this ordinary formalism of the quantum theory of fields the theory is divided into two distinct sections: the section giving the kinematical relations between various quantities at the same instant of time, and the section determining the causal relations between quantities at different instants of time. Thus the commutation relations belong to the first section and the Schr?dinger equation to the second.

‘As stated before, this way of separating the theory into two sections is very unrelativistic, since here the concept “same instant of time” plays a distinct role.

‘Also in our formalism the theory is divided into two sections, but now the separation is introduced in another place. One section gives the laws of behaviour of the fields when they are left alone, and the other gives the laws determining the deviation from this behavior due to interactions. This way of separating the theory can be carried out relativistically.

‘Although in this way the theory can be brought into more satisfactory form, no new contents are added thereby. So, the well-known divergence difficulties of the theory are inherited also by our theory. Indeed, our fundamental equations (8.17) admit only catastrophic solutions, as can be seen directly from the fact that the unavoidable infinity due to non-vanishing zero-point amplitudes of the fields inheres in the operator *H* _{12}(*P*). Thus, a more profound modification of the theory is required in order to remove this fundamental difficulty.

‘It is expected that such a modification of the theory could possibly be introduced by some revision of the concept of interaction, because we meet no such difficulty when we deal with the non-interacting fields. The revision would then have the result that in the separation of the theory into two sections, one for free fields and one for interaction, some uncertainty would be introduced. (p.272) This seems to be implied by the very fact that, when we formulate the quantum field theory in a relativistically satisfactory manner, this way of separation has revealed itself as the fundamental element of the theory.’

So, although Tomonaga indeed discovered the Tomonaga-Schwinger equation first, he was in 1943 still far from seeing how to resolve the fundamental problems of the theory. In particular, he believed that the solution lay in additional interactions. In contrast, Schwinger’s conservative bent five years later led to the insistence on retaining the electromagnetic interaction, with the divergences absorbed by the process of renormalization.

As we recall, in 1948, Oppenheimer arranged to have a brief note published in the *Physical Review*, summarizing the progress in quantum electrodynamics which had occurred in Japan since the end of the war.^{20} This note expressed the reaction to the news* of the experimental discovery of the Lamb shift, and Bethe’s^{6} and Schwinger’s [43] theoretical contributions. Tomonaga first reported on his group’s unsuccessful attempt^{41} to use the method of compensation.^{42} Then, after seeing the work of Bethe,^{6} they were able to absorb infinities into a reinterpretation of the mass and charge of the electron, i.e. renormalization of these physical quantities. However, they made an error, and found additional divergences in the *e* ^{2} correction to the Klein-Nishina formula for Compton scattering. As Oppenheimer remarked in an attached comment, ‘From manuscripts kindly sent by Tomonaga, I would conclude that the difficulties referred to in this note result from an insufficiently cautious treatment, and therefore inadequate identification, of light quantum self-energies.’ The letter concludes with a statement that a calculation of the Lamb shift was in progress (by Yoichiro Nambu), which included the anomalous magnetic moment effect found by Schwinger [43]. By September 1948 Tomonaga’s group had reproduced the correct relativistic Lamb shift calculation of French and Weisskopf,^{23} albeit using non-covariant techniques.^{43} The paper appeared in 1949, as did Schwinger’s [53] and Feynman’s^{26} papers on the Lamb shift.

# Tomonaga after the war

As we have seen, Oppenheimer was very impressed with Tomonaga’s accomplishments in wartime Japan, and invited him to spend a year at the Institute for Advanced Study. This he did during the academic year 1949-50. When he returned to Japan, he soon abandoned scientific work for the role of science administrator. In part, this was because of Nishina’s death in January 1951, so Tomonaga became the chairman of the Liaison Committee for Nuclear Research. He was responsible for the establishment of various national research institutes, such as Yukawa Memorial Hall, renamed first the Research Institute for Fundamental Physics, and then the Yukawa Institute for Theoretical Physics, (p.273) in Kyoto. He was evidently a skillful administrator, known for the ‘Tomonaga method of arbitration’ making ‘decisions when everyone gets tired.’ Yet, there was a great loss to physics when this genius stopped making original contributions to fundamental knowledge.

Tomonaga was modest about his contributions. In a letter he wrote from Princeton in 1950 to Ziro Koba, ‘At the end of last year, I listened to Dr. J. Schwinger’s lecture at Columbia.* He also seems to have a new idea. It is a very ambitious plan to put Dr. F. J. Dyson’s argument in a closed form without a series expansion in powers of *e* ^{2}/*ħc*. (I hear that Toichiro Kinoshita, too, has the same ambition. I myself wanted to do the same and struggled long and hard, but in vain. I am disgusted with my lack of progress….) Anyway, the three, J. Schwinger, R. Feynman and F. J. Dyson, are great men, and I must admit defeat. (I have not met Dyson yet—he, too, might come to Princeton next school year, I hear.) People refer to the Tomonaga-Schwinger theory or the Schwinger–Tomonaga theory (especially in lapan), but the comparison of the two may be likened to the one between H. A. Lorentz and G. F. FitzGerald of the Lorentz-FitzGerald contraction (no need to tell which one corresponds to Lorentz and which to FitzGerald). I hear that my name has appeared in various magazines and newspapers following the awarding of the Nobel Prize to Hideki Yukawa, and I find it very embarrassing.’^{35}

## Schwinger’s view of Tomonaga

Ultimately, Schwinger did not feel that the work of Tomonaga made a significant impact, at least on his own work. ‘I had nothing to learn from what Tomonaga and his group had done, because when it came to things like the Lamb shift, they were just dutifully following previous success. I have no doubt that they were still using a subtraction theory—it was not satisfactory. It was done in a pseudo-physical context by having a spin-zero particle or something that would produce a negative mass change that would cancel the positive mass change, which does nothing for vacuum polarization. Oppenheimer then pointed out in response to this shipment of papers that whatever [Tomonaga] was doing had no effect on
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the divergences, so-called, in the photon mass.’^{1} Writing down the Tomonaga-Schwinger equation was not the major step: ‘I’d like to make the point that from a covariant formulation to a covariant calculation is a big step. It seems to me the formulation was trivial. It was carrying through the calculation that was the important thing. The thing that surprises me is that so many people refer to this 1943 paper as anticipating the whole line of development of renormalization theory and so on. That is a formalism without any physical content. The idea of renormalization is a very specific strategy of isolating parts of the result, identifying them as being altered properties of the individual particles, and going on. To point to a vacuous covariant theory and say that’s the whole thing is patently wrong.’^{1} This feeling of paucity of the Japanese contribution at the time would present Schwinger with difficulties years later when he would deliver a memorial lecture in Tokyo in honor of Tomonaga. Nevertheless, he did deliver a moving tribute, which we will describe in Chapter 16.

In his history of his development of quantum electrodynamics, Schwinger elaborated on this point: ‘I have read remarks to the effect that if scientific contact had not been broken during the Pacific war, the theory that we are reviewing here would have been significantly advanced. Of course, lacking an unlimited number of parallel universes in which to act out all possible scenarios, such statements are meaningless. Nevertheless, I shall be bold enough to disagree. The preoccupation of the majority of involved physicists was not with analyzing and carefully applying the known relativistic theory of coupled electron and electromagnetic fields but with changing it. The work of Tomonaga and his collaborators, immediately after the war, centered about the idea of compensation, the introduction of the fields of unknown particles in such a way as to cancel the divergences produced by the known interactions.^{41} Richard P. Feynman also advocated modifying the theory, and he would later intimate that a particular, satisfactory modification could be found.^{26} My point is merely this: A formalism such as the covariant Schr?dinger equation is but a shell awaiting the substance of a guiding physical principle. And the specific concept of the structure-independent renormalized relativistic electrodynamics, while always abstractly conceivable, in fact required the impetus of experiments to show that electrodynamic effects were neither infinite nor zero, but finite and small, and demanded understanding.’ [199]

# Feynman’s theory of positrons, and the space-time approach to quantum electrodynamics

Much has been written about Feynman’s scientific accomplishments, both at the popular level,^{44} and from the scholarly point of view. For the latter, we refer the reader to Schweber’s book^{2} and to Mehra’s biography.^{45} As we will
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see, Feynman’s approach to quantum electrodynamics seemed to be totally different from that of Schwinger and Tomonaga, or indeed from that of any of the field theorists of the 1930s and 1940s. His approach was far more intuitive (to him at least), less mathematical (on the surface anyway), and apparently revolutionary (as opposed to Schwinger’s conservative road); yet remarkably, as both Feynman and Schwinger came to realize in 1948, the two procedures were equivalent. Freeman Dyson proved that equivalence in 1949.

Feynman started on his unorthodox path at Princeton, while working on his PhD with John Archibald Wheeler. Already, while he was an undergraduate at MIT, he was concerned with the infinities of electrodynamics, in particular the infinite self-action of the electron on itself. Perhaps, he thought, one could just impose a rule that a given electron does not interact with itself. But that could not be correct, because radiation reaction, which must be present to preserve the energy balance between the electron and the electromagnetic field, would then not occur either. Feynman and Wheeler got the idea that the self-action could be eliminated by making what seemed like an outrageous change in the boundary conditions of ordinary classical electrodynamics: Instead of having only retarded waves, in which the waves reach the observer from the past, they proposed having a classical electrodynamics in which one had half-retarded and half-advanced waves, waves which come from the future. This had the theoretical advantage of being time-symmetric, that is, invariant under the change of the sense of the flow of time, from past to future, to future to past, so that the boundary conditions in time mirror the symmetry in Maxwell’s equations. It was not quite as simple as that, in that perfectly absorbing boundaries had to be assumed as well. But then radiation reaction could be accounted for, as Feynman noted later: ‘It became clear that there was the possibility that if we assume all actions are via half-advanced and half-retarded solutions of Maxwell’s equations and assume that the sources are surrounded by material absorbing all the light which is emitted, then we could account for radiation resistance as direct action of the absorber acting back by the advanced waves on the source.’^{46}

Wheeler proposed that Feynman give a colloquium in the fall of 1940 at Princeton on their joint work, and Wheeler would follow later with a colloquium on the corresponding quantum theory, which Wheeler, but not Feynman, thought would be an easy generalization. Feynman was terrified, because Pauli would be in the audience, but Wheeler promised to take care of all of Pauli’s questions. Although Pauli indeed asked questions, no one in attendance could, in later years, recall them. Einstein remarked that it would be difficult to follow the same path in gravitation theory, but since that was a much less well-established theory, that was not a serious argument against the approach.^{45}’^{2} Later, in February 1941, Feynman gave a talk on time-symmetric electrodynamics at a meeting of the American Physical Society in Cambridge.^{47}
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Only after the war did the Wheeler-Feynman paper^{48} finally appear, a long paper written almost entirely by Wheeler. It described only classical electrodynamics, as an action-at-a-distance theory. Each charged particle was the source of an advanced and a retarded field, which only acted on other particles. Only the particles are fundamental entities. ‘From the overall space-time view of the least action principle, the field disappears as nothing but bookkeeping variables insisted on by the Hamiltonian method.’^{46}

Early in the collaboration between Wheeler and Feynman, an idea occurred to Wheeler that would be very important for Feynman’s later thinking about quantum electrodynamics. The question was why do all electrons possess the same mass and charge. ‘Because,’ said Wheeler in 1940, ‘they are all one and the same electron.’^{46}’^{49} By this, Wheeler meant that there was only one worldline of an electron, which zig-zagged, sometimes going forward in time, in which case it was an electron, and sometimes going backwards in time, in which case we saw it as a positron, with the same mass as the electron, but with the opposite charge. Feynman doubted there was but one such electron in the world (if so, the number of electrons and positrons would seem to have to be the same, manifestly in contradiction to experience), but very much liked the idea that a positron was merely an electron going backward in time. It seemed a much more attractive idea than Dirac’s holes in a filled electron sea. This notion would play a crucial role in Feynman’s diagrammatic interpretation of quantum electrodynamics at the end of the decade. (See Fig. 8.1 below.)

The next step in Feynman’s journey was the principle of least action. The action, for a single classical particle with coordinate *q*(*t*), is given by the integral

*t*

_{1}and

*t*

_{2}are the initial and final times, and

*L*is the Lagrangian of the system. The classical stationary action principle states that the trajectory of the particle is such that the action

*S*is an extremum, which yields the Lagrange equation,

These equations may be immediately extended to a system described by an arbitrary number of generalized coordinates, *q* _{α}.

Feynman’s inspiration for the quantum theory, as had Schwinger’s, came from Dirac. In this case it was his paper ‘The Lagrangian in quantum mechanics,’^{50} a paper which would be the springboard for Schwinger’s later
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action-principle-based field theory, which we will describe in the next chapter.* In that paper Dirac stated that the transformation function in the coordinate representation between two different times, (*x* _{t2}∣*x* _{tl}), ‘is *analogous* to’ exp [(i/ħ)S(*x* _{2}, *t* _{2}; *x* _{1}, t_{1})], the exponential factor being the action carrying a particle from an initial position *x* _{1} at time *t* _{1} to a final position *x* _{2} at time *t* _{2}. No one, including Dirac, seemed to know what ‘analogous to’ meant in this case. Perhaps, Feynman thought in 1941, that it meant ‘equal [or, rather, proportional] to.’ Thus was born Feynman’s famous path integral.

In fact the transformation function and e^{iS/ħ} were proportional if the time interval were short, *t* _{2} *– t* _{1} ≪ *t* _{1}. To calculate the transformation function *K*(X, *T; x, t*) that carries one from a wavefunction ψ(*x, t*) to a wavefunction ψ(*X, T*) required breaking up the interval into a great many steps, say AT, and integrating over each intermediate position:

*N*goes to infinity, at the same time as the size of all the time intervals goes to zero; in that sense it resembles the definition of a Reimann integral.

In 1942, Feynman wrote up his PhD thesis which consisted of the work on the new approach to quantum mechanics and the action-at-a-distance electrodynamics. (These were only fully described after the war, the first in the joint paper with Wheeler,^{48} and the second in an article by himself, entitled ‘Space-time approach to nonrelativistic quantum mechanics.’^{51}) He then spent his full time on war work, and soon, after his marriage to Arline Greenbaum, departed
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for Los Alamos, where he was placed in charge of Theoretical Computations. It would not be until he was well settled as a professor at Cornell, in 1946, that he would again resume fundamental research. But his debt to his advisor, John Wheeler, with his tremendous geometrical way of thinking, was incalculable, for it would lead to Feynman’s space-time view of electrodynamics.

## Feynman after Shelter Island

Like Schwinger, Feynman was excited by the experimental results announced at the Shelter Island Conference in June 1947. He set to work, and by the time of the Pocono Conference the following March, he, like Schwinger, had a relativistically invariant computational scheme. We have described Feynman’s presentation at Pocono in the last chapter. But that conference belonged to Schwinger, and Feynman’s unconventional approach was not received with much favor. He realized that only through publication could he hope to convince the community that he was on the right track.

As we described in Chapter 7, at the January 1948 APS meeting in New York, after Schwinger’s famous repeated lecture on the anomalous magnetic moment and the preliminary unsatisfactory situation with the relativistic Lamb shift calculation, Feynman got up and stated that he agreed with Schwinger’s results, but he, unlike Schwinger, had the correct value of the anomalous magnetic moment for an electron in the atom. (Actually, the discrepancy was with the corresponding electrical coupling obtained from the magnetic one by a relativistic transformation.) He was at that time feeling a tremendous sense of competition with Schwinger, who had got a head start on him, but now Feynman felt, probably overconfidently, that he had caught up.^{2}’^{45}

Feynman published two relatively short papers bearing on this subject in the summer of 1948. The first was entitled ‘A relativistic cut-off for classical electrodynamics,’^{52} which was based on an expanded version of a manuscript he had written in 1941.^{45}’^{2} This paper dealt largely with the action-at-a-distance formulation he worked on before getting involved in the war effort, but now with a density of field quanta playing the role of a regulator, so that the self-energy of a particle was made finite. A similar idea was present in the second paper, ‘Relativistic cut-off for quantum electrodynamics.’^{22} He used this to calculate the self-energy of the electron, (μ = electron mass)

_{0}is a cutoff, which in conventional electrodynamics would tend to infinity. This is the result (8.13), first obtained in the old quantum field theory by Weisskopf,

^{12}published by Schwinger five months later [52]. In fact, this paper directly precedes Schwinger’s ‘Quantum electrodynamics

*V*in the

*Physical*(p.279)

*Review*, which was received by the journal about two weeks after Feynman’s paper. Since it uses old-fashioned methods, which Feynman used in part to make it acceptable to other physicists,

^{2}this paper is mainly remembered for its incorrect discussion of the relativistic Lamb shift, which we will describe below.

The moment when Feynman achieved confidence in the power of his methods came at the January 1949 APS meeting in New York. This is the famous story of Murray Slotnick, who had spent six months calculating a certain interaction between electrons and neutrons using either a pseudoscalar or a pseudovector interaction. The first form gave a finite result, while the second was divergent. After his talk, Oppenheimer challenged Slotnick: ‘What about Case’s theorem’ Ken Case, a former student of Schwinger’s who was then a postdoc at the Institute for Advanced Study, had a proof that the pseudovector and pseudoscalar theories were equivalent. Feynman was intrigued, so he talked to Slotnick, and that evening he worked out the general result for arbitrary momentum transfer. When he talked to Slotnick the next day, Feynman found that Slotnick only had the result for zero momentum transfer. But in that limit they agreed. Feynman was ecstatic: ‘That was the moment when I got my Nobel prize, when Slotnick told me he had been working for two years. When I got the real prize, it was really nothing, because I already knew I was a success.’^{2} Later, after he learned the meaning of creation and annihilation operators, Feynman found the error in Case’s theorem.^{2}’^{45}’^{53}

Feynman’s substantial papers on quantum electrodynamics appeared in 1949. These were ‘The theory of positrons,’^{54} received by *Physical Review* on 8 April 1949, and ‘Space-time approach to quantum electrodynamics,’^{26} received a month later. The validity of the rules given in these two papers was demonstrated in a third paper, ‘Mathematical Formulation of the Quantum Theory of Electromagnetic Interactions,’^{55} which arrived at *Physical Review* over a year later, on 8 June 1950. (All three of these papers are reprinted in Schwinger’s collection [83].)

‘The theory of positrons’ is summarized in the abstract. ‘The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms.

‘In this solution, the “negative energy states” appear in a form which may be pictured (as by Stückelberg^{56}) in space–time as waves traveling away from the
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external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the electron. A particle moving forward in time (electron) in a potential may be scattered forward in time (ordinary scattering) or backward (pair annihilation). When moving backward (positron) it may be scattered backward in time (positron scattering) or forward (pair production). For such a particle the amplitude for transition from an initial to a final state is analyzed to any order in the potential by considering it to undergo a sequence of such scatterings.

‘The amplitude for a process involving many such particles is the product of transition amplitudes for each particle. The exclusion principle requires that antisymmetric combinations of amplitudes be chosen for those complete processes which differ only by exchange of particles. It seems that a consistent interpretation is only possible if the exclusion principle is adopted. The exclusion principle need not be taken into account in intermediate states. Vacuum problems do not arise for charges which do not interact with one another, but these are analyzed nevertheless in anticipation of application to quantum electrodynamics.

‘The results are also expressed in momentum-energy variables. Equivalence to the second quantization theory of holes is proved in an appendix.’

Feynman began by considering a classical picture of pair production, followed by positron annihilation. An electron-positron pair is produced at time *t* _{1}, after which two world lines, corresponding to the electron and positron, advance forward in time. At some later time *t* _{2} the positron is annihilated by another electron. The picture might be as sketched in Fig. 8.1. As he said, ‘Following the charge rather than the particles corresponds to considering this continuous world line as a whole rather than breaking it up into pieces. It is as though a bombardier flying low over a road suddenly sees three roads and it is only when
(p.281)
two of them come together and disappear that he realizes that he has simply passed over a long switchback in a single road.’*

Feynman went on to consider the Green’s function for Schrodinger’s equation, which he defines as relating the wavefunction at two different space-time points:

He proceeded to solve the Dirac equation for a particle of mass *m* in an external potential *A* _{μ} (here he used the notation A = *y* _{μ}A_{μ}, ∇ = *y* _{μ}∂_{μ})

Here Feynman had adopted a compressed notation, in which the numbers 2 and 1 stand for the space-time coordinates with the respective index. It is clear that this differential equation is equivalent to the integral equation

*A*= 0. The subscript here refers to the appropriate boundary conditions in time. In order that Feynman’s theory be equivalent to the hole theory, he had to choose the free Green’s function so that it involved a sum over positive energy states for positive time differences, and a sum over negative energy states for negative time differences:

Here (φ_{n} is an eigenfunction of the free Dirac Hamiltonian, with energy *E* _{ny} and ${\overline{\phi}}_{n}={\phi}_{n}^{*}\beta $ is the Dirac conjugate.

(p.282)
Quantum electrodynamics proper is the subject of the second paper, ‘Space-time approach to quantum electrodynamics.’^{26} * The first paragraph of the abstract gives a good summary: ‘In this paper, two things are done. (1) It is shown that a considerable simplification can be attained by writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons.’

In this paper, Feynman gives the famous Feynman rules and the Feynman diagrams. These may be illustrated in the momentum-space diagram, representing the ‘interaction of an electron with itself,’ shown in Fig. 8.2. This diagram has a precise mathematical correspondence with a quantum mechanical amplitude, in this case, the divergent integral,

Indicated in the figure are the various factors that are assembled in order to construct the amplitude (8.27).

As he stated, Feynman’s second step was a modification of electrodynamics so that these integrals would be rendered convergent. He does this, in effect, by modifying the photon propagator l/k^{2} by multiplying it with a convergence
(p.283)
factor *C*(*k* ^{2}) which falls off at least as fast as l/k^{2}, so now integrals such as that in Eqn (8.27) converge. For example, we could take C(k^{2}) = λ^{2}/(λ^{2} *– k* ^{2}), which tends to unity as λ→ ∞. (Actually, Feynman proposed averaging over a weight function G(λ), with the property${\int}_{o}^{\infty}{\lambda}^{2}G(\lambda )d\lambda =0$ Doing so for the case of the process here, given by Eqn (8.27), gave a result for the electron mass shift exactly of the form (8.21), as given first by Feynman and then by Schwinger the year before.

Feynman next considered radiative corrections to scattering, in particular the Lamb shift. There he admitted the error he had previously published in the ‘Relativistic cut-off for quantum electrodynamics.’^{22} The story is recounted in his famous footnote 13: ‘That the result given in B^{22} was in error was repeatedly pointed out to the author, in private communication, by V.F. Weisskopf and J.B. French, as their calculation, completed simultaneously with the author’s early in 1948, gave a different result. French has finally shown that although the expression for the radiationless scattering … is correct, it was incorrectly joined onto Bethe’s nonrelativistic result. He shows that the relation In 2*k* _{max} – 1 = ln λ_{min} used by the author should have been ln 2*k* _{max} – 5/6 = In λ_{min}. This results in adding a –1/6 to the logarithm in B, Eqn. (8.19), so that the result now agrees with that of J.B. French and V.F. Weisskopf^{23} and N.M. Kroll and W.E. Lamb.^{24} The author feels unhappily responsible for the very considerable
(p.284)
delay in the publication of French’s result occasioned by this error. This footnote is appropriately numbered.’*

However, Feynman faced a real difficulty with vacuum polarization. His ‘regularization’ scheme did nothing to remove the divergence associated with a closed electron loop, as given by the amplitude

^{†}as did Feynman. He realized that in the hole theory they were necessary for probability conservation. He suggested that the Lamb shift measurement be sufficiently improved so that the vacuum polarization contribution, which amounted to –27 MHz compared to a total splitting of 1050 MHz, could be experimentally confirmed.

He did finally discuss a method of regularizing vacuum polarization which he attributed (without reference) to Bethe and Pauli. This evidently was the Pauli-Villars technique,^{21} which Feynman call ‘the superposition of the effects of quanta of various masses (some contributing negatively).’ This gave rise to a renormalization of the charge, again depending logarithmically on a cutoff λ,

Feynman closed the paper by discussing spin-0 particles, and meson theories in this language. This was a payoff from the Slotnick episode. He was able to reproduce all sorts of meson-theoretic calculations using his rules to order *g* ^{2}
(p.285)
very easily, much to his delight. But comparison with experiment was not very fruitful, because of the largeness of the coupling.

Feynman’s paper on the ‘Mathematical formulation of the quantum theory of electromagnetic interaction’^{55} was designed to justify the space-time procedure given in the previous papers, and supply the ‘proof of the equivalence of these rules to the conventional electrodynamics.’ (In fact, the first four sections of this paper were written in 1947, much of which duplicated the work in Feynman’s thesis.^{2}) It was followed a year later by ‘An operator calculus having applications in quantum electrodynamics,’^{58} which was completed while Feynman was on leave of absence in Brazil, before taking up his new permanent appointment at Caltech. It is important to note Feynman’s leisurely publication schedule. As Feynman said, ‘Dates don’t mean anything. It was published in 1951, but it had all been invented by 1948.’^{59}

This last paper remains of some interest.*
Feynman began by discussing the ordering of operators, in particular the meaning of e^{A+B} when *A* and *B are* non-commuting. The question is, how is this ‘disentangled’ into its dependence on the individual operators, for only if A and *B* commute is it equal to e^{A}e^{B}. This is a subject for which the work of Schwinger is justly famous.^{†} Feynman went on to apply his calculus to quantum mechanics, in particular to a system coupled to a harmonic oscillator, and to field theory, quantum electrodynamics in particular. He supplied his own derivation of the Tomonaga-Schwinger equation (8.2). He used his procedure to supply ‘an independent deduction of all the main formal results in quantum electrodynamics, by use of the operator notation.’ He then rederived the quantum-mechanical amplitudes for processes he had computed by his intuitive technique in Ref. 26.

These papers completed Feynman’s program in quantum electrodynamics. ‘With this paper I had completed the project on quantum electrodynamics. I didn’t have anything remaining that required publishing. In these two papers^{55,58} I put everything I had done and thought should be published on the subject. And that was the end of my published work in the field.’^{59}

Feynman left the field of quantum electrodynamics in triumph, but personally he was dissatisfied. He thought that he would solve the problem of the divergences in the theory, that he would ‘fix’ the problem, but he didn’t. ‘I invented a better way to figure, but I hadn’t fixed what I wanted to fix. I had
(p.286)
kept the relativistic invariance under control and everything was nice … but I hadn’t fixed anything…. The problem was still how to make the theory finite…. I wasn’t satisfied at all.’^{2} In fact, in ‘Space-time approach to quantum electrodynamics’ he apologized for not having solved the problem: ‘The desire to make the methods of simplifying the calculation of quantum electrodynamic processes more widely available has prompted this publication before an analysis of the correct form for the [cutoff function] *f*+ is complete.’^{26} He was also disappointed that his space–time picture of electrodynamics wasn’t really new, that it was, in fact, equivalent to the conventional field theory of Schwinger and Tomonaga. He had hoped to eliminate fields entirely as fundamental entities in favor of particles, but field theory had triumphed in the end.

## Schwinger’s perspective

Let us conclude this section by giving Schwinger’s perspective on Feynman’s contributions to the development of quantum electrodynamics, extracted from the Preface to *Quantum electrodynamics* [83]. Referring to his own line of attack he stated: ‘Throughout these developments the basic view of electromagnetism was that originated by Maxwell and Lorentz—the interaction between charges is propagated through the field by local action. In its quantum-mechanical transcription it leads to formalisms in which charged particles and fields appear on the same footing dynamically. But another approach is also familiar classically; the field produced by arbitrarily moving charges can be evaluated, and the dynamical problem reformulated as the purely mechanical one of particles interacting with each other, and themselves, through a propagated action at a distance. The transference of this line of thought into quantum language^{54}’^{26}’^{55} was accompanied by another shift in emphasis relative to the previously described work. In the latter, the effect on the particles of the coupling with the electromagnetic field was expressed by additional energy terms which could then be used to evaluate energy displacements in bound states, or to compute corrections to scattering cross-sections. Now the fundamental viewpoint was that of scattering, and in its approximate versions led to a detailed space-time description of the various interaction mechanisms. The two approaches are equivalent; the formal integration of the differential equations of one method supplying the starting point of the other.^{61} But if one excludes the consideration of bound states, it is possible to expand the elements of a scattering matrix in powers of the coupling constant, and examine the effects of charge and mass renormalization, term by term, to indefinitely high powers. It appeared that, for any process, the coefficient of each power in the renormalized coupling constant was completely finite.^{57} This highly satisfactory result did not mean, however, that the act of renormalization had, in itself, produced a more correct theory. The convergence of the power series is not established, and the series doubtless has the significance of an asymptotic
(p.287)
expansion. Yet, for practical purposes, in which the smallness of the coupling constant is relevant, this analysis gave assurance that calculations of arbitrary precision could be performed.’

# Dyson and the equivalence of the radiation theories of Schwinger, Tomonaga, and Feynman

As we have mentioned, already in 1948 (although the proof was only published in 1950^{55}) Feynman had proved, to his satisfaction, the equivalence of his space-time approach to quantum electrodynamics, and the more conventional, yet equally brilliant, canonical approach of Schwinger. But Feynman never received the credit for this demonstration, largely because of his slow publication schedule. In fact, as we have just seen, it is invariably Freeman Dyson who is credited with proving the equivalence of the two, seemingly very different, approaches to quantum field theory.

We have recounted Dyson’s interactions with Schwinger and Feynman in the previous chapter. When Bethe showed Dyson the letter Tomonaga had written to Oppenheimer, Dyson was delighted, for he found Tomonaga’s exposition transparent, whereas the notes from Schwinger’s lectures at Pocono seemed complicated, and penetrable only by the master himself.^{2} Dyson attended the Michigan lectures of Schwinger in the summer of 1948, finding them ‘unbelievably complicated.’ Dyson felt Schwinger’s approach ‘couldn’t be the way to do it,’ for it was ‘something that needed such skills that nobody besides Schwinger could do it. If you listened to the lectures you couldn’t see the motivation; it was all hidden in this wonderful apparatus.’^{2} In contrast, by this time he was already on very friendly terms with Feynman, with whom he had driven across the country. So before he took up his new residence in Princeton, he had already established his allegiance.

He saw early on, perhaps more explicitly than did either Feynman or Schwinger, the connection between the two methodologies. What is remarkable is that he published his papers, ‘The radiation theories of Tomonaga, Schwinger, and Feynman,’^{61} and ‘The *S* matrix in quantum electrodynamics,’^{57} received by *Physical Review* on 6 October 1948 and 14 February 1949, well before Feynman’s central paper, ‘The theory of positrons,’^{54} received on 8 April 1949. Moreover, the first appeared before Schwinger’s ‘QED If [52], which established the divergence structure of the theory, and both before ‘QED III,’ Schwinger’s definitive paper of the triad. It could be argued that Dyson’s alacrity in publication ensured his place in history, whereas had he published after the principals had completed their expositions, his contributions would have appeared more minor.

In his first paper, Dyson started from the Tomonaga-Schwinger equation (8.2), which makes reference to the interaction representation. He then gave a perturbative solution to that equation for the time-evolution operator in (p.288) powers of the interaction Hamiltonian. This expansion is, in general, only possible for that part of the interaction referring to the coupling of matter to the radiation field, given by Eqn. (8.3). He then went on to contrast, and relate, the approaches of Schwinger and Feynman. The former is characterized by an operator which ‘represents the interaction of a physical particle with an external field, including radiative corrections,’ which may be expressed in terms of ‘characteristic’ repeated commutators:

Here H^{1} is the interaction Hamiltonian (8.3) with a mass shift term removed,

^{e}is the remaining part of the interaction Hamiltonian, for example, the interaction to the Coulomb field of the nucleus. In Dyson’s perhaps critical words, ‘The repeated commutators in this formula are characteristic of the Schwinger theory, and their evaluation gives rise to long and rather difficult analysis.’* (In a note added in proof, Dyson noted he had given an incorrect interpretation of Schwinger’s formulation, and in fact Schwinger’s approach, like Feynman’s, was symmetric between past and future.) But Dyson’s main point here was not an explication of Schwinger’s methods, but of Feynman’s.

Dyson’s key innovation was the introduction of a time ordering operator *P*. ‘If

*x*

_{1}, …,

*x*

_{n}of space-time, then

*x*

_{1}), …, σ(

*x*

_{n}) occur in time.’ The Feynman theory was then seen to be given in terms of a time-ordered product of interaction (p.289) operators:

Dyson went on to calculate matrix elements. In so doing, he used his time-ordering notation to define the ‘Feynman’ propagators for the photon and the electron:

*x, y*) is ±1 depending on whether σ (

*x*) is later than or earlier than σ (

*y*), and the subscripts on the fermion fields are Dirac indices. In terms of these, Dyson was able to derive Feynman’s graphical rules.

Note that in fact Dyson had made a major break with Feynman, who insisted on the particle nature of electrons, while Dyson, like Schwinger and Tomonaga, saw everything as fields. ‘Nobody at Cornell understood that the electron field was a field like the Maxwell field. That was something that was in Wentzel^{62} but was nowhere else. That was what was lacking in the old fashioned way of calculating. The electron was a particle, the photon was a field, and the two were just totally different. This notion of just two interacting fields with the simple interaction term ψ̇γ_{μ}A_{μ}ψ was essentially what I brought to Cornell with me from England out of Wentzel’s book.’^{62,2}

This paper appeared shortly after Dyson assumed his visiting fellowship at the Institute for Advanced Study, whose director was J. Robert Oppenheimer. Dyson was invited to present several seminars on this work. Oppenheimer, although initially expressing interest, was very hostile until Bethe intervened; then Oppenheimer capitulated and became a believer. But Dyson was not happy with him: ‘Oppenheimer was a great disappointment. He hadn’t time for the details. As compared to Hans Bethe, Oppenheimer was completely superficial. To talk to Oppenheimer was interesting. It was like meeting some very famous person who had interesting things to say but I just never got anything that you could really call guidance. I wasn’t needing much guidance…. He had a bad effect on other people who needed the guidance more than I did.’^{2} These remarks are not dissimilar to those of Schwinger concerning his interactions with Oppenheimer in Berkeley a decade earlier.

It was the second paper of Dyson, ‘The S matrix in quantum electrodynamics’^{57} that assured his fame. In this paper he recast Schwinger’s and
(p.290)
Feynman’s electrodynamics into what has become the standard form. As Dyson stated in the introduction, ‘The present paper deals with the relation between the Schwinger and Feynman theories when the restriction to one-electron problems is removed. In these more general circumstances, the two theories appear as complementary rather than identical. The Feynman method is essentially a set of rules for the calculation of elements of the Heisenberg *S* matrix corresponding to any physical process, and can be applied with directness to all kinds of scattering problems. The Schwinger method evaluates radiative corrections by exhibiting them as extra terms appearing in the Schr?dinger equation of a system of particles and is suited especially to bound-state problems. In spite of the difference of principle, the two methods in practice involve the calculation of closely related expressions; moreover, the theory underlying them is in all cases the same. The systematic technique of Feynman, the exposition of which occupied the second half of I^{61} and occupies the major part of the present paper, is therefore now available for the evaluation not only of the *S* matrix, but also of most of the operators occurring in the Schwinger theory.’

Dyson gave a systematic exposition of the perturbation theory of quantum electrodynamics. He did so by giving the so-called Schwinger-Dyson equations. These consisted of an infinite set of coupled integral equations for the Green’s functions of the theory. For example, the full electron and photon propagators, *S*'_{F}, *D*'_{F}, satisfied by the equations

_{v}(

*k, k*+

*q*) is a vertex amplitude coupling a vector potential

*A*

_{v}(

*q*), corresponding to a photon with momentum

*q*to incoming and outgoing electrons with momenta

*k*and

*k*+

*q*, respectively, which in turn is determined by still further integral equations. The perturbative solution to this system of (p.291) equations, where in lowest order Γ

_{v}= γ

_{v}, leads to Feynman’s rules for the construction of all quantum mechanical amplitudes for computing scattering processes in QED.*

Dyson concluded his paper by discussing renormalization. He showed that if the propagators and the vertices were multiplied by certain constants,

_{i}could be so chosen as to cancel the divergences occurring in perturbation theory, and the resulting Green’s functions of the theory were entirely finite. The finite renormalized charge

*e*was given in terms of the bare charge

*e*

_{0}by

*e*= ${z}_{3}^{1/2}{e}_{0}$

Dyson continued his contributions to field theory with a series of major papers published in 1951, dealing with what he called Heisenberg operators. This was somewhat in the spirit of Schwinger’s canonical transformation designed to isolate the renormalization effects, but unlike Schwinger, Dyson did not use the adiabatic (slowly varying) approximation. He gave an exposition of this program at the Michigan Summer School in 1950. When the papers were published the following year^{64} Dyson felt that he had made a major contribution that would ‘get radiation theory moving forward again’ and would allow the application of field-theory methods to meson problems. Unfortunately for Dyson, more effective methods rapidly became available, including Schwinger’s Green’s function techniques [66], so these papers had negligible impact at the time.^{2}’^{†}

A concluding paper published by Dyson in 1952 had significant repercussions on the view of the meaning of perturbation theory in quantum field theory.
(p.292)
This was ‘Divergence of perturbation theory in quantum electrodynamics.’^{65} There he gave a simple argument that perturbation theory could not result in a convergent series. The argument went as follows: suppose one computed a Green’s function as a series in powers of *e* ^{2} (Apart from an overall factor, any Green’s function has an expansion in powers of *e* ^{2} or α.) If the series were convergent for sufficiently small values of *e* ^{2} it would have to converge even if *e* ^{2} were small but negative. But this cannot be, for if *e* ^{2} were negative like charges would attract, and the vacuum would be unstable to decay into an arbitrarily large number of electron-positron pairs. At best then, perturbation theory must result in an asymptotic series, which nowhere converges, but for which a finite number of terms gives an optimal approximation to the true Green’s function. This is not an obstacle in practice for quantum electrodynamics, since the coupling constant, α = 1/137, is so small. But the proof was discouraging to Dyson: ‘That was, of course, a terrible blow to all my hopes. It really meant that this whole program [of perturbative quantum field theory] made no sense.’^{2} Nowadays, no one is seriously disturbed about the asymptotic nature of perturbation theory, although it does raise the unresolved issue of the importance of non-perturbative effects in field theories, be they quantum electrodynamics or quantum chromodynamics (the theory of strong interactions). There is also the beginning of a recognition that Dyson’s argument may be wrong, because it fails to take into account boundary conditions.^{66}

## The impact of Dyson’s work

The predominant view of the impact of Dyson’s work was beautifully given by C.N. Yang. ‘The papers of Tomonaga, Schwinger, and Feynman did not complete the renormalization program since they confined themselves to low-order calculations. It was Dyson who dared to face the problem of high orders and brought the program to completion. In two magnificently penetrating papers, he pointed out and resolved the main problems of this very difficult analysis. Renormalization is a program that converts additive subtractions into multiplicative renormalization. That it works required a highly non-trivial proof. That proof Dyson supplied. He defined the concepts of primitive divergences, skeleton graphs, and overlapping divergences. Using these concepts, he pushed through an incisive analysis and completed the proof of renormalizability of quantum electrodynamics. His perception and power were dazzling.’^{67}

But the inventors of renormalized quantum electrodynamics were less impressed. In a later interview, Schwinger expressed his view of the contributions of Dyson to quantum electrodynamics. He began by paraphrasing Feynman: ‘“Of course, neither you nor I needed to be told that our theories were equivalent and we didn’t need Dyson.” And, of course, that was true. Dyson was writing not for us, but for the rest of the world. What Dyson contributed
(p.293)
was … the utility of a formal construction of that unitary operator in terms of time-ordering. There is the point that Dyson recognized that Feynman throughout was always dealing with scattering problems, that his theory in principle was incapable of dealing with bound states. Dyson recognized that I had a more complete theory. It was a Hamiltonian theory; you could deal with energy eigenvalues and so forth. Dyson did contribute something in his recognition of the importance of the time-ordering formulation. And that is what underlay the particular propagation function that Feynman and, as we know, Stückel-berg before him, had introduced. From a practical point of view, I think he was simply translating his understanding of what Feynman was trying to do—and it’s not clear that Feynman would necessarily have agreed with all that—into the ordinary language of operators and so forth. And pointing out that the different handling of the operators would produce the Feynman result. Valuable. Not world-shaking, but valuable.’^{1}

In fact, as we have noted, Schwinger reacted positively to Dyson’s introduction of time-ordering, recognizing its superiority: ‘If you look at my own work you will see not time-ordering, but a concern with symmetrical and anti-symmetrical products. Therefore, two functions. Whereas the complex time-ordered [propagation] function ultimately turns out to be the more convenient thing.’^{1} In Schwinger’s view it was fortunate that Dyson’s paper^{61} was published before Feynman’s.^{54} ‘Feynman’s paper published by itself would probably never have communicated very well. Dyson recognized what quantum-mechanical formulation Feynman was implicitly using, which was very valuable because nobody else could possibly have understood it without that recognition.’ Dyson’s papers were useful ‘as one of the gospels, the interpretation of the mystical words to the masses.’^{1} But Schwinger was unhappy at the success of the Feynman-Dyson approach: ‘I confess it utterly astonished me that his method became so popular. That, of course, was not Feynman’s doing but Dyson’s. Without Dyson using my language to translate Feynman it never would have been understood.’^{1}

Feynman, perhaps, had more cause for unhappiness, because Dyson’s papers appeared before his. For a while, people even talked about ‘Dyson graphs.’ But Feynman was not too concerned. In later remarks, he commented, ‘He wasn’t trying to steal anything from me; he hadn’t claimed they were his. All he was trying to do was tell everyone that there was something good in my theory, that he had discovered the connection with the work of Tomonaga and Schwinger, and that all these different approaches were equivalent. This greatly helped people to understand the different theories. His paper had some crazy language which I couldn’t understand, but others could understand it. It was like a translation of my theory, my language, for other people; of course, it’s a mistake to translate something for the author. I was bothered only slightly, and (p.294) I would be more concerned today if they were still called “Dyson graphs”. That would not make me miserable, but I would complain a little bit about it.

‘A little later, the diagrams came to be called “Dyson-Feynman graphs”, with some others calling them the “Feynman graphs” through a number of people who knew about their origin a little better. Now, of course, it is as it should be. “We write down *the* diagram for this or that process.” And that’s the best, because it’s anonymous, its *the* diagram. It makes me feel better than the “Feynman diagram”, because it is *the* rule for something, and that’s just fine.’^{59}

# Feynman and Schwinger—cross-fertilization

Although Schwinger and Feynman never collaborated, and talked together rather rarely, it is clear that there was a certain synergism between these two innovators who nearly simultaneously scaled the peak of electrodynamics.

They had of course rather different goals: Schwinger was interested in understanding the experimental situation. ‘I was concentrating on understanding these electromagnetic phenomena. I developed a formalism adequate enough to account for it, period. Feynman had something more grandiose in mind from the very beginning, a reconstruction of quantum mechanics using more intuitive ideas, and these same electromagnetic problems were for him simply a way of understanding what he was trying to do. These particular problems were not the center of his interest as they were for me. They were just another bit of experimental data in order to evolve his ideas. So Feynman was aiming at a more general method to begin with, but he could not have gotten there without the concrete answers, shall I say, that I provided and which he could then adapt and on the basis of which put forward his more general method. I don’t know quite how to say it except that his aim was ultimately more far-reaching, but he needed—we were complementing each other. We were not in competition. Our ambitions were different. I got to these answers very quickly, which rather contradicts the general opinion that I used very complicated incomprehensible methods. They went fast and I don’t ascribe it to any particular talents that I have. The machinery was perfectly okay for the purpose that it was being invented for. Whereas Feynman was looking for something more general.’^{1}

Feynman indeed influenced Schwinger to find a better method to work out higher-order effects. ‘Let’s face it, the method that I had got clumsier and clumsier. Any method does at higher order. Perhaps a little more rapidly, which is why, when I finally realized what Feynman was trying to do, I took a look at it and went back and found a more general method myself. Which is perfectly reasonable. I’m emphasizing the point that what I did was more than adequate for the limited questions being asked, it explained the Lamb shift and the magnetic moment to the accuracy at which they were then measured. When
(p.295)
the accuracy increased, the theory had to go to higher orders. Then came the question of which way of formulating it was most efficient and something along Feynman’s line was no question [more efficient] and I adapted myself to it’ albeit with a differential rather than an integral attitude.^{1} In the year 1948–49 ‘I was certainly deeply involved in trying to look for more general formulations and seeing what there was in the Feynman–Dyson things that I should have to adopt, to find a synthesis. These were clearly not so different paths, but variations on each other. [The question was] what ultimately was the best version. I spent a lot of time on that. Particularly looking at all kinds of higher-order effects, for example, the two-particle differential equation, which became known as the Bethe-Salpeter equation, which I was talking about a year earlier and describing in lectures at Harvard. That was certainly the future, not the past.’^{1}

Schwinger expressed regret that his interactions with Feynman had not been stronger. ‘We were kind of moving in similar directions. It’s too bad we couldn’t have interacted earlier. We could have saved the world a lot of time. If he had gone to Columbia, we would have worked together at a much earlier stage. The reformulation of quantum mechanics might have occurred earlier and then that would have vastly simplified the application to electrodynamics.’^{1}

References

Bibliography references:

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(p.296)
15. Letter of Dyson F. J. to his parents, dated 4 February 1948,quoted in Ref. 2, p. 320. 16. W. E. Lamb, Jr, and R. C. Retherford, *Phys. Rev.* 72, 241 (1947).

17. See footnote 5 of [53].

18. J. R. Oppenheimer, talk at *Les Particules Élémentaires*, Rapports et Discussions du Huitième Conseil de Physique tenu a l’Universite libre de Bruxelles du 27 Septem-bre au 14 Octobre 1948, R. Stoops, Brussels, 1950, reprinted in *Quantum electrodynamics* [83] as paper number 15 (p. 145), and quoted in Jagdish Mehra, *The Solvay conferences on physics*, Reidel, Dordrecht, Holland and Boston, USA, pp. 257–259. 19. S. Tomonaga, *Prog. Theor. Phys.* 1, 27 (1946). 20. S. Tomonaga, *Phys. Rev.* 74, 224 (1948). 21. W. Pauli and F. Villars, *Rev. Mod. Phys.* 21, 434 (1949). 22. R. P. Feynman, *Phys. Rev.* 74, 1430 (1948). 23. J. B. French and V. F. Weisskopf, *Phys. Rev.* 75, 338 (1949). 24. N. M. Kroll and W. E. Lamb, Jr, *Phys. Rev.* 75, 388 (1949). 25. J. M. Luttinger, *Phys. Rev.* 74, 893 (1948). 26. R. P. Feynman, *Phys. Rev.* 76, 769 (1949). 27. E. A. Uehling, *Phys. Rev.* 48, 55 (1935). 28. G. Wentzel, *Phys. Rev.* 74,1070 (1948). 29. Oppenheimer Papers, quoted in Ref. 2, p. 351. 30. K. Tanaka, conversation with K. A. Milton, Vancouver, British Columbia, 28 July 1998. 31. R. E. Marshak, ‘From two mesons and (*V– A*) weak currents to the standard model of quark and lepton interactions,’ lectures given at the University of Rochester, October 1987, VPI-HEP-87/7, UR-1041. 32. S. Sakata and T. Inoue, *Prog. Theor. Phys.* 1,143 (1946). 33. S. Tomonaga, *Bull. I. P. C. T.* (*Riken-iho*) 22, 545 (1943). 34. F. J. Dyson, *Disturbing the Universe*. Harper & Row, New York, 1979, p. 57. 35. M. Matsui (ed.), *Sin-itiro Tomonaga*—*life of a Japanese physicist* [English version edited and annotated by H. Ezawa, translated by C. Fujimoto and T. Sano]. MYU, Tokyo, 1995. 36. S. Tomonaga, ‘My childhood in Tokyo’ in Ref. 35, pp. 26–27. 37. Y. Tomonaga, ‘My brother and his childhood environment’ in Ref. 35, pp. 40–43. 38. W. Heisenberg, *Die Physikalische Prinzipien der Quantentheorie*. Hirzel, 1930. 39. P. A. M. Dirac, *Proc. Roy Soc. London* 136, 453 (1932). 40. E. Stückelberg, *Helv. Phys. Acta* 11, 225 (1938). 41. D. Ito, Z. Koba and S. Tomonaga, *Prog. Theor. Phys.* 2, 216 (1947); 3, 276 (1948). 42. A. Pais, *Phys. Rev.* 68, 227 (1946); S. Sakata, *Prog. Theor. Phys*. 2, 30 (1947). 43. K. Fukuda, Y. Miyamoto and S. Tomonaga, *Prog. Theor. Phys.* 4,47,121 (1949). 44. J. Gleick, *Genius: the life and science of Richard Feynman*. Vintage Books, New York, 1992.
(p.297)
45. Jagdish Mehra, *The beat of a different drum: the life and science of Richard Feynman*. Claredon Press, Oxford, 1994. 46. R. J. Feynman, Nobel Lecture, 11 December 1965, *Science* 53, 699 (1966), p. 2. 47. R. P. Feynman and J. A. Wheeler, *Bull Am. Phys. Soc.* 16, 683 (1941). 48. J. A. Wheeler and R. P. Feynman, *Rev. Mod. Phys.* 17, 157 (1945). 49. R. P. Feynman, interview by Jagdish Mehra, January 1988. 50. P. A. M. Dirac, *Phys. Z. Sowjetunion* 3(1), 64 (1933). 51. R. P. Feynman, *Rev. Mod. Phys.* 20, 367 (1948). 52. R. P. Feynman, *Phys. Rev.* 74, 939 (1948). 53. K. M. Case, *Phys. Rev.* 75,1506 (1949); 76,14 (1949). 54. R. P. Feynman, *Phys. Rev.* 76, 749 (1949). 55. R. P. Feynman, *Phys. Rev.* 80, 440 (1950). 56. E. C. C. Stuckelberg, *Helv. Phys. Acta* 15, 23 (1942). 57. F. J. Dyson, *Phys. Rev. 75*, 1736 (1949). 58. R. P. Feynman, *Phys. Rev.* 84, 108 (1951). 59. R. P. Feynman, conversations and interviews with Jagdish Mehra, 1970, 1988, and interviews with Charles Weiner, American Institute of Physics, 1966. 60. *Science Citation Index*, lISI, Philadelphia, 1998. 61. F. J. Dyson, *Phys. Rev.* 75, 486 (1949). 62. G. Wentzel, *Einführung in der Quantentheorie der Wellenfelder*. Franz Deuticke, Vienna, 1943 [English transl: *Quantum theory of fields*. Interscience, New York, 1949]. 63. Roy Glauber, interview with K. A. Milton, in Cambridge, Massachusetts, 8 June 1999. 64. F. J. Dyson, *Phys. Rev.* 82, 428, 608 (1951); 83, 1207 (1951); *Proc. Roy. Soc. London* A207, 395 (1951). 65. F. J. Dyson,*Phys. Rev.* 85, 631 (1952). 66. C. M. Bender and K.A. Milton, *J. Phys. A* 32, L87 (1999). 67. C. N. Yang, *Selected Papers, 1946–1980, with Commentary*. W. H. Freeman, San Francisco, 1983, p. 65.

## Notes:

(*)
At that meeting, which was held at George Washington University, Schwinger recalled doing clandestine calculations, in lieu of note-taking, using hydrogenic wavefunctions to understand the large value of the Bethe logarithm in the Lamb shift, obtaining an estimate within about 10% of the exact value [199]. Here is another version of the story we reported in the previous chapter. ‘It was a rather small group as I remember, sitting around a table. I must, at the same time, have been thinking not only about the magnetic moment, which I had carried through to the point where I could see there was an answer, but there were a few integrals to do, and so forth, but I was also thinking about what Bethe had done—you know there’s a logarithm involving summation over the energy levels of hydrogen and Bethe said numerically the answer is this, and I wasn’t very satisfied. While I was listening to lectures at George Washington University of no great interest to me, listening to people speculate about the universe and other such things, I was sitting there and doing a little scribbling and calculation of my own, to get a qualitative feeling for how that number came out. I was astonished that Bethe didn’t actually do the numbers, because he was perfectly capable of doing it.’^{1}

(*)
The formula was unfortunately mistyped by the journal to read (½Π) *e* ^{2}/*ħc*, which was copied in Rosenfeld’s book.^{3}

(†)
Schwinger was slightly upset by Bethe’s publication. We have recounted in the previous chapter how Weisskopf was annoyed with Bethe for single-handedly taking credit for this result. ‘It struck me roughly the same way, but not quite as forcibly as it struck Weisskopf, who I think has been quoted as being rather angry at Bethe’s so rapidly stealing the thunder. Because the essence of it was what Weisskopf and I talked about and I think we had a somewhat different version of it, that if one calculated the two energy levels, the *S* and *P* levels, and looked at their difference, all the ultraviolet divergences would cancel and one would end up with a finite result. I was not personally upset about it, because to me the challenge was the relativistic calculation, which Bethe did not touch. Clearly a large part of the Lamb shift was nonrelativistic so my interest shifted to what was clearly a totally relativistic effect, the magnetic moment.’^{1}

(*)
Recall that his student notebooks at City College contained detailed notes of major papers on field theory by Dirac, Heisenberg, and Pauli and Weisskopf from the 1920s and 1930s.^{8}

(*)
‘Kramers wrote a book on quantum mechanics^{11} in which he goes through some pedestrian development and, I believe, points out the infinite self-energy and then says that clearly we have quantized the wrong classical theory. The correct classical theory should already have removed from it this deficiency of classical electromagnetism, namely the infinite mass of a point charge. And when you corrected the classical theory, then that is the proper thing to quantize.’ But this approach to mass renormalization does not work, ‘Because you cannot find a classical theory on which you can superimpose phenomena like pair creation and other things which are necessary and part of the relativistic quantum electrodynamics. It is a dead end. Nevertheless, it looks superficially as though Kramers invented mass renormalization.’^{1}

(†) Schwinger noted that ‘it was W. Furry who first appreciated the logarithmic nature of the divergence of the electromagnetic mass in the hole theory of electrons and positrons.’ [199]

(*)
The only entry in the list of abstracts for the meeting under Schwinger’s name was that for a paper presented with Weisskopf ‘On the electromagnetic shift of energy levels’ [45], which was a description of Schwinger’s first, flawed, relativistic treatment of the Lamb shift. This was not based on a collaboration between Weisskopf and Schwinger, but merely on discussions the two shared over lunches in a good French restaurant. After some time, the atmosphere was spoiled because crowds of students started tagging along, as we described in Chapter 5. Schwinger simply stopped coming.^{1}

(*)
This may have occurred earlier, in September, when he first calculated an invariant electromagnetic mass shift in his original non-covariant approach.^{1}

(†)
Schwinger recalled that Rabi was somehow involved in bringing the work of the Japanese to his attention. ‘Rabi was in Japan and obviously talked to the important Japanese physicists and must have brought back papers of what they were working on.’^{1}

(*) Schwinger later remarked about his repeated APS lecture at Columbia in January 1948: ‘The only record I have of that event is a typed copy of my already submitted report [43], on the back page of which is written a formula for the energy shift of hydrogenic levels. One of the terms is a spin-orbit coupling, which should be the relativistic electric counterpart of the α/27π additional magnetic-moment effect. But it is smaller by a factor of 3; relativistic invariance is violated in the non-covariant theory…. But the back of the page also contains something else—the answer to the obvious question: What happens if the additional magnetic-moment coupling to the electric field is given its right value, no other change being introduced? What emerges, and therefore was known in January 1948, is precisely what other workers using non-covariant methods would later find, which is also the result eventually produced by the covariant methods. Of course, until those covariant methods were developed and applied, there could be no real conviction that the right answer had been found.’ [199] As we noted in the previous chapter, Feynman, at that meeting, announced that he had no such difficulty in obtaining the correct magnetic moment effect using his own, totally different, but covariant, technique. But Feynman apparently did not have a value for the Lamb shift at that time—and his subsequent covariant result, like Schwinger’s, was erroneous.

(*)
In fact, in the Appendix to Feynman’s paper in which he introduces a parameter integral to combine denominators, he states that it was ‘suggested by some work of Schwinger’s involving Gaussian integrals.’^{26} As Schwinger stated, ‘The technique of invariant parameters was the technique that Feynman borrowed from me.’^{1}

(*)
Tanaka, who became a professor at The Ohio State University, recalled that he was one of the first group of Japanese students sent to the US after the war, and served as courier from Tomonaga to Oppenheimer.^{30} Marshak^{31} recalled that Tanaka delivered the Sakata-Inoue paper,^{32} which suggested the existence of two mesons (see Chapter 12), to Oppenheimer in November 1947, and presumably the QED papers as well.

(*) Years later, because of this uncle’s actions, Tomonaga would be unable to attend the Nobel Prize ceremony in Stockholm. After learning of his nephew being awarded the prize, Masuzo visited with a bottle of sake, and after drinking for some hours, Tomonaga slipped in the bath, breaking six ribs, making traveling impossible. But he always recalled this incident with affection: ‘You see, my uncle came around first thing in the morning with a bottle of sake, and ….’

(*) The educational system, modeled on Germany’s, at the high school level consisted of five schools, numbered First through Fifth, located in Tokyo, Sendai, Kyoto, Kanazawa, and Kumamoto, respectively. The B in the department designation indicated that the first foreign language was German, rather than English or French.

(*)
Through a report in *Newsweek*.

(*)
In a December 1949 letter to Tatuoki Miyazima, he says about Schwinger’s lecture: ‘His lucid talk was very impressive. Until I heard his lecture, I thought him to be adept only in steamroller-tactic calculations, and not so sharp. However, I realized that he was not such a man as I had imagined, but one who was working with a very clear concept of physics. His lecture was on renormalization—how we can renormalize the mass and the electric charge in closed form. Dyson subtracted infinity by using a series expansion in powers of α(= e^{2}/*ħc*). Therefore, his method is good only for collision problems, but not appropriate for dealing with the bound state. Schwinger tried ambitiously to subtract infinity in a way appropriate for bound-state problems, and to reduce the subtraction to renormalizations of mass and charge. Of course, I was able to understand only the basic gist of his thinking.’^{35}

(*)
Schwinger would later remark, ‘Dirac was central to this in the connection between quantum mechanics and classical mechanics, shall we say. Action in general. There are two different ways of looking at it. Feynman picked up the integral aspect of it in which you combine little steps in time into an integral formulation. I picked up another remark in that very same paper, namely the differential aspect, the quantum aspects and analogies with Hamilton-Jacobi and so forth. So ultimately to the extent that we finally diverged with attitudes about reformulations of quantum mechanics—which is what I think this is all really about—we were both inspired by Dirac, but took two different avenues, which are equivalent in limited contexts. I like to think that the differential aspect is more fundamental, because it is not based on mimicking of a classical situation. If everything is classical, then what do you do about non-classical degrees of freedom, like Fermi-Dirac fields and spins and such things. Whereas the differential aspect allows both possibilities, it is not so confining in the nature of the system to which it refers.’^{1}

(*)
In an interview with Schweber, Feynman stated that this metaphor ‘was suggested to me by some student at Cornell (who had actually been a bombardier during the war) when I was writing up the paper and was asking for opinions of how to explain it and only had poor or awkward metaphors.’^{2}

(*)
In a remarkable demonstration of how close the competition was between Feynman and Schwinger, this paper appeared in the *Physical Review* directly before Schwinger’s ‘QED III,’ which was received exactly 17 days later, on 26 May 1949. Recall that Feynman’s ‘Relativistic cut-off in quantum electrodynamics’ had also appeared directly before Schwinger’s ‘QED I,’ which again was received by the journal exactly 17 days after Feynman’s paper, on 29 July 1948.

(*)
According to footnote8 in an earlier-published paper of Dyson,^{57} it was Schwinger who detected the incorrect use of the insertion of a ‘photon mass’ to match the high-energy with the low-energy contributions to the Lamb shift.

(†)
Schwinger remarked: ‘Vacuum polarization means no more than that an electron-positron combination is coupled to the electromagnetic field and it may show itself really or virtually as you like.’ Schwinger knew that vacuum polarization was real from his work with Oppenheimer at Berkeley [15]. And his work on classical electrodynamics was invaluable: as with the resistive and reactive parts in synchrotron radiation, ‘the overtly physical and the implicitly physical parts are all connected together, you don’t keep one and throw the other one away. In other words, I had lots of preparation in other areas of physics. I’m not sure Feynman did. He was too abstract.’^{1}

(*)
According to the *Science Citation Index*,^{60} this paper had a very respectable 19 citations in 1997 alone.

(†)
This general problem was discussed in an appendix to a paper Schwinger wrote with Robert Karplus, with the unlikely title of ‘A note on saturation in microwave spectroscopy’ [44], received by *Physical Review* on 9 January 1948. We mentioned this paper briefly in Chapter 5.

(*)
To which Schwinger responded: ‘Well, it wasn’t so long and it wasn’t so difficult, but nevertheless it was not the most economical way of going on to higher order effects. That I not only grant, but I insist on…. He did recognize that, as I think Feynman probably didn’t, that the Feynman theory does operate with a statement about initial and final states, which is a concentration on the overall evolution of the system. And that was a useful thing. No question about it. And as soon as I understood that, I immediately incorporated it into my own next version as well.’^{1}

(*) This equation appears explicitly in Schwinger’s 1951 paper, ‘On the Green’s function of quantized fields’ [66].

(*)
Glauber recounted an embarrassing error that Schwinger made in this connection. In fall 1949 Schwinger gave a long sequence of lectures at the joint theoretical seminar hosted by Harvard and MIT on the Green’s functions of quantum electrodynamics; in effect he claimed to have found a closed integral expression for the vertex function Γμ. John Blatt took notes of these seminars, and they reached Norman Kroll at Columbia, who discovered a crucial error: the scattering of light by light had been inadvertently omitted. Shortly thereafter, Pauli visited Harvard from the Institute for Advanced Study, having heard of this error from Kroll, and visited Schwinger in his office. Sometime later, Schwinger emerged, ‘badly shaken: Pauli was delighted to be the bearer of bad news.’ Of course, in those early days, the structure of field theory was poorly glimpsed, so it is understandable that such an error could escape even the master.^{63}

(†)
These four papers of Dyson’s had no citations in 1997 according to the *Science Citation Index*.^{60}