Quantum electrodynamics in the 1930s

Quantum electrodynamics or QED is said to be the quantum field theory or QFT which gives “a complete account of matter and light interaction”. Some say it was developed by Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman in the 1940s:

Image from Rod Nave’s hyperphysics

But some say it started with Pascual Jordan in 1925, some say it started with Dirac in 1927, and some say it started with Heisenberg and Pauli’s “canonical” papers of 1929 and 1930. In the history of QFT Meinard Kuhlmann says quantum electrodynamics was “the historical as well as systematical entrée” to quantum field theory, and Heisenberg and Pauli established the basic structure. So it sounds as if QED as we know it started in 1930, when QED and QFT were the self-same thing. Silvan Schweber talked about it in his Scholarpedia article quantum field theory: origins. Well worth a read.

The complete failure of the theory

The 1930s were troubled times, and QED was troubled too. The Wikipedia history of quantum field theory article says it was “plagued by several serious theoretical difficulties”, and the situation was dire, desperate, and gloomy. The problem of infinities or “divergence” was the big one. On page 185 of his 1997 book conceptual development of 20th Century field theories, Tian Yu Cao says it stemmed from classical electrodynamics. He refers to Henri Poincaré’s 1906 paper on the dynamics of the electron, and to Enrico Fermi’s 1922 observation that the Poincaré electron is not stable. The idea was that it would take infinite energy to push two like-charged point-particles together, so the same must apply when you tried to assemble an electron. Hence it would have infinite self-energy. Cao says this elicited a response which was first stated by Yakov Frenkel in 1925. Frenkel said electrons “have no extension in space at all. Inner forces between the elements of an electron do not exist because such elements are not available”.

The point-particle electron was quickly accepted

Cao tells how this point-particle electron was quickly accepted by physicists and became the “conceptual basis of the ideas of local excitations and local interactions in QFT”. On page 85 of his 1993 book dreams of a final theory, Steven Weinberg says the resultant problem was noticed by Heisenberg and Pauli and by Ivar Waller, but appeared most clearly and disturbingly in Julius Robert Oppenheimer’s 1930 note on the theory of the interaction of field and matter. Oppenheimer said “the theory, is, however, wrong, since it gives a displacement of the spectral lines… which is in general infinite”. In 1931 Lev Landau and Rudolf Peierls wrote their extension of the uncertainty principle to relativistic quantum theory. They talked of absurd results and the complete failure of the theory, and said “it would be surprising if the formalism bore any resemblance to reality”. On page 690 of the Oxford companion to the history of modern science, Silvan Schweber said the problem caused most of the workers in the field to doubt the correctness of QFT, and the many proposals advanced in the 1930s all ended in failure. This is somewhat surprising given that the Davisson-Germer experiment and the Thomson and Reid diffraction experiments demonstrated the wave nature of matter, not the point-particle nature of matter.

Dirac didn’t care about the problem of infinities

Heisenberg responded to Oppenheimer in 1930, but with a paper on the self-energy of electrons which mused about a minimum-length lattice world, then about a theory where the electron mass was zero. Meanwhile Dirac had come up with a theory of electrons and protons. That’s where space is some infinite Dirac sea, and consists of an infinite number of negative-energy electrons per unit volume, some of which have infinite negative energy. All these negative-energy electrons are of course completely unobservable, just like the angels on the head of the pin. Then in 1931 Dirac wrote quantised singularities in the electromagnetic field which further suggests he didn’t care much about the problem of infinities. This is where he’s said to have predicted the positron. However it feels like a leap too far from holes to positrons. It feels like Dirac was moving the goalposts after Oppenheimer’s paper on the theory of electrons and protons. And it feels like Dirac was lucky to get credit for predicting the positron. Graham Farmelo talks about it in his 2010 article did Dirac predict the positron? He says Dirac’s “close friend Patrick Blackett, one of the leading players in the story’s denouement, denied it”. And that “very few physicists took Dirac’s hole theory seriously” . He also says “Victor Weisskopf later recalled the idea ‘seemed incredible and unnatural to everybody’”.

Fermi’s 1932 review article

Perhaps that’s why in his 2002 article Enrico Fermi and quantum electrodynamics 1929 – 1932, Silvan Schweber said a generation of physicists learned quantum electrodynamics from Fermi’s 1932 article quantum theory of radiation. He also said “none of the alternative formulations – Werner Heisenberg and Wolfgang Pauli’s, in particular – combined the simplicity, transparency, and thoroughness of Fermi’s approach”. Fermi started with an analogy of a pendulum and a vibrating string connected by a thin elastic thread. He talked about standing waves in a cavity and electromagnetic energy in space and the spinning electron, then gave five ways to apply the theory. Whilst the mathematics is arguably heavy and the physics is arguably light, there is a clarity to Fermi’s writing. Part 2 is perhaps less interesting because it’s about Dirac’s theory of the electron, but at least Fermi was critical of hole theory. Part 3 is quantum electrodynamics, where he said no charged-particle field could be represented as a superposition of electromagnetic waves. This suggests he’d never made a Möbius strip. But never mind, he was critical of Heisenberg and Pauli’s method, saying he preferred to use “the method of the writer, which is more simple and more analogous to the methods used in the theory of radiation”. He ended up talking about meeting “a very serious difficulty, since the electrostatic energy of point charges is infinite”. He said no satisfactory answer had been given, and that “in conclusion we may therefore say that practically all the problems in radiation theory which do not involve the structure of the electron have their satisfactory explanation; while the problems connected with the internal properties of the electron are still very far from their solution”. Sadly Fermi’s attempt at a theory of β rays was rejected by Nature. So he switched from theoretical physics to experimental physics, and nobody followed up on the structure of the electron.

Dirac talks of messenger photons

Dirac certainly didn’t. in his 1932 paper on relativistic quantum mechanics you can read that classical electrodynamics “teaches us that the idea of an interaction energy between particles is only an approximation and should be replaced by the idea of each particle emitting waves, which travel outward with a finite velocity and influence the other particles in passing over them”. It simply isn’t true. An electron doesn’t go round and round in a magnetic field because it’s being influenced by waves moving at the speed of light. Hence one raises an eyebrow when Dirac said “we must find a way of taking over this new information into the quantum theory”. Later he said “let us suppose we have a single electron interacting with a field of radiation and consider the radiation resolved into ingoing and outgoing waves”. That’s fair enough for Compton scattering, where an E=hf photon interacts with an electron. The interaction is genuinely caused by a photon moving from A to B at c. But Dirac used this to claim that all electromagnetic interactions are caused by photons moving from A to B at c. It’s another leap too far, one that ignores Compton’s 1921 paper the magnetic electron as well as electron spin and the hard scientific evidence:

Image from Quantum Electrodynamics, The interaction of light with matter by Tristan Roddis

It feels like Dirac was winging it with fantasy physics, but somehow got away with it. Other work by Dirac includes his 1932 paper the Lagrangian in quantum mechanics. It’s on page 312 of Julian Schwinger’s selected papers of quantum electrodynamics. On page 29 you can find Dirac’s 1932 paper on quantum electrodynamics co-authored with Vladimir Fock and Boris Podolsky. Part I is a proof that Heisenberg-Pauli theory was equivalent to the Dirac theory, and part II is the Maxwellian case where they talk about the introduction of a separate time for the field and for each particle. That sounds like fantasy physics too. Helge Kragh says something about it on page 137 of his 1990 book Dirac: a scientific biography. He says Pauli considered it a great improvement.

1933 and the theory is still wrong

In 1933 Dirac wrote his Solvay paper on the theory of the positron, where he was still talking about holes. He said “one might think at first that with such an infinity the whole theory becomes untenable”. Then he swept it under the carpet with an arbitrary cutoff. He also wrote a paper called a statement of a problem in quantum mechanics. He said “the limitations of the new theory appear when one tries to get a formulation conforming to restricted relativity”. He wasn’t talking about the problem of infinities. Others were. In their April 1933 paper on the question of the measurability of electromagnetic field quantities, Léon Rosenfeld and Niels Bohr said the “difficulties disappear if one uses test bodies whose linear extensions are chosen sufficiently large”. However they didn’t mention Schrödinger, or Darwin, or pair production. Even though Carl Anderson discovered the positron in August 1932, and Patrick Blackett and Giuseppe Occhialini discovered pair production in March 1933. Surely by now everybody knew about the wave nature of matter, and that electrons and positrons were made out of light? Why were they still talking about point particles? Why did Rosenfeld and Bohr say “the fluctuations in question are intimately related to the impossibility… of visualizing the concept of light quanta in terms of classical concepts”. There is no thought here about what a photon is, or what an electron is, or what happens in pair production. Or what a positron is. Or why when it goes round and round in a magnetic field, it goes the other way.

A patch-up job for the point-particle electron

And so the problems rumbled on. See page 187 of Cao’s conceptual development of 20th Century field theories. He says in 1933 Gregor Wentzel introduced a “limiting process” into the definition of the Lorentz force, and that the problem of self-energy was first studied most thoroughly by Victor Weisskopf. There’s an English version of Weisskopf’s 1934 paper on the self-energy of electrons on page 157 of Miller’s early quantum electrodynamics. Weisskopf was Pauli’s assistant. He said the Pauli exclusion principle prevents two electrons being found closer together than approximately one de Broglie wavelength. Hence “one will find at the position of the electron a ‘hole’ in the distribution of the vacuum electrons”. But one can also find around the electron “a cloud of higher charge density coming from the displaced electrons”. And thus “there is a broadcasting of the charge of the electron over a region of the order h/mc”. Plus “the electric field energy of an electron in QFT is negative”. There is no understanding of charge here. On page 188 Cao says this result was one of the starting points for the theory of mass renormalization. That might be true, but I’m afraid it demonstrates no understanding of the electron. The field isn’t the electric field or the magnetic field, it’s the electromagnetic field, and the field energy is positive.

Creation and annihilation operators are no substitute for understanding

Moving swiftly on, on page 175 Cao refers to the 1934 paper on the theory of the electron and positive by Wendell Furry and Oppenheimer. They talked of wavefunction and pair production, but said nothing about waves of very great curvature. Cao says the paper mitigated the idea that the vacuum was the scene of wild activities in which infinite-negative-energy electrons existed. And that “the filled-vacuum assumption could be abandoned”. But it was not abandoned. Cao also says “the same method of exchanging the creation and destruction operators for negative states was used in the same year by Pauli and Weisskopf”. But creation and annihilation operators are no substitute for understanding what happens in gamma-gamma pair production. As per Gregory Breit and John Wheeler’s 1934 paper on the collision of two light quanta, you start with two waves moving linearly at c, and you end up with two “spinor” particles. With a wave nature. Which are not moving linearly at c. How difficult can it be to hazard a guess as to what happened there? Unfortunately the 1934 paper by Pauli and Weisskopf was on the quantization of the scalar relativistic wave equations. It was their “anti-Dirac” paper, but it talked about charged particles without spin, which totally misses the trick. There’s an English translation on page 188 of Miller’s early quantum electrodynamics.

The subtraction physics

On page 58 Miller talks about Dirac’s 1934 paper discussion of the infinite distribution of electrons in the theory of the positron. Dirac said “we now have a picture of the world in which there are an infinite number of negative-energy electrons (in fact an infinite number per unit volume) having energies extending continuously from -mc² to –∞”. No we don’t. We now have a picture of fantasy physics totally at odds with experimental evidence, electromagnetism, and wave mechanics. Dirac also said “the problem now presents itself of finding some natural way of removing infinities”. How about not introducing them in the first place? Dirac went on to talk about a “distribution R which can be divided naturally into two parts R = Ra + Rb where Ra contains all the singularities”. That doesn’t feel natural at all. On page 60 Miller describes how an immediate response to Dirac’s paper came from Peierls, who at the time happened to be working with Dirac. See Peierls’ paper the vacuum in Dirac’s theory of the positive electron. He said this: “According to a theory proposed by Dirac one has to picture the vacuum as filled with an infinite number of electrons of negative kinetic energy, the electric density of which is however unobservable”. He also said the discovery of the positive electron gives strong support to this view, when it absolutely does not. Rather ominously Miller says Peierls succinctly put the goal of what Pauli would call “the subtraction physics”, and that statements closer and closer to modern renormalization theory are beginning to appear.

Heisenberg has a go at subtraction physics

Miller gives a run-down of Weisskopf’s 1934 paper on the self-energy of electrons, then on page 61 says Heisenberg had other thoughts. He quotes Heisenberg, in a letter to Bohr, saying a point charge appears analogous to a quantum mechanical system in the deepest quantum state. And that “we must promote in the quantum mechanics of waves that a single point charge is a trivial solution of the equations of motion”. It would seem as if Heisenberg had no concept of waves of very great curvature. It’s as if Schrödinger had never been born. Miller also says Heisenberg, in a letter to Pauli, calculated the charge density and obtained a result wherein “a polarization of the vacuum is no longer left over”. And that Heisenberg wrote a letter to Pauli and Weisskopf saying “no polarization of the vacuum is necessarily correct and moreover trivial”. Miller says Pauli generally approved, and Weisskopf wrote back saying the polarization of the vacuum should be finite but not zero.

Vacuum polarization image from the standard model of electroweak interactions by Antonio Pich

Miller also quotes Weisskopf saying Pauli’s disgust was only against the subtraction physics, not the problem itself. On page 64 Miller refers to Heisenberg’s 1934 subtraction-process paper remarks on the Dirac theory of positron. I’m afraid to say it feels like another magical mathematical physics-free zone. Heisenberg said his purpose was “to construct the Dirac theory of the positron in the formalism of quantum electrodynamics”. He talked about a quantum-mechanical system of many electrons characterized by a density matrix from which physically-important properties like charge density could be “read off”. On page 10 he said “we first assume that a scalar potential A₀, which is regarded as a small perturbation, is slowly introduced and then kept constant, and then ask what sort of matter is created by it from originally empty space”. He had no idea how matter is created. On page 19 he said “a light quantum generates a matter field in its neighborhood in a manner that is similar to the way that an electron generates a Maxwell field”. No it doesn’t. He had no concept of what a photon is, or an electron. Section I was the “intuitive theory of matter waves” but there was no intuition. Yes, there are glimmers of hope. Such as on page 19 where he referred to Halpern and Debye and the scattering of light by light “even when the energy of the light quanta is not sufficient for pair creation”. But he hardly mentioned spin, and he said “the calculations in these problems are so complicated that they will not be attempted here”. He did say “a pure separation of the fields that are involved into matter fields and electromagnetic fields is scarcely possible any more”. And that a unified theory of matter and light fields will make possible a contradiction-free union of quantum theory with intuitive field theory. But Miller tells us Pauli was critical, and that Heisenberg said the entire subtraction physics contains nonsense and should be replaced by something better.

Pauli had thrown out the baby but kept the dirty bathwater

Talking of which, on page 193 of their book climbing the mountain: the scientific biography of Julian Schwinger, Jagdish Mehra and Kimball Milton say Pauli proposed a solution for the infinite charge density problem. It involved taking equal portions of the electron sea and the positron sea. Then the resulting charge of the vacuum will be zero, since “the vacuum charge of the electron sea will be exactly compensated by the vacuum charge of the positron sea”. Talk about doubling down on fantasy physics. Mehra and Milton also say this: “in the resulting improved theory, not only was the picture of the vacuum simple again, but there were now only three fundamental interactions between the electrons, the positrons, and the photons: the scattering of the fermion with the emission or absorption of the photon, and the annihilation or creation of the electron–positron pair with the emission or absorption of the photon”. It would seem that the interaction between photons had gone, and that Pauli had thrown out the baby but kept the dirty bathwater.

Stueckelberg’s covariant perturbation theory circumvents infinities

It wasn’t the only time that happened. Ernst Stueckelberg came up with his covariant perturbation theory in 1934. He was the gilded youth who’d had a great start in physics, but ended up unlucky and unwell. See Stueckelberg’s biography on the St Andrews website by John O’Connor and Edmund Robertson. They say he gave an explanation of nuclear interactions in terms of vector boson exchange, but didn’t publish because Pauli said it was ridiculous, leaving the field free for Hideki Yukawa. Stueckelberg’s covariant paper was Relativistisch invariante Störungstheorie des Diracschen Elektrons (Relativistic invariant perturbation theory of the Dirac electron). For details see The Road to Stueckelberg’s Covariant Perturbation Theory by Jan Lacki, Henri Ruegg, and Valentine Telegdi. They say Stueckelberg’s contribution “constitutes the first complete and easily generalizable instance of a manifest relativistically invariant perturbative calculus”. In section 6 they say Stueckelberg thanked Wentzel for the suggestion to work on an invariant perturbation theory. They also say its modernity is striking and the approach was powerful, but it was not adopted by others at the time, and the lack of interest was unfortunate. They quote Weisskopf from 1981 saying this: “already in 1934 it seemed that a systematic theory could be developed in which these infinities are circumvented. At that time nobody attempted to formulate such a theory. There was one tragic exception, and that was Ernst C G Stueckelberg”. Lacki et al say Stueckelberg’s expressions for matrix elements are identical to those obtained nowadays from Feynman diagrams. And that Pauli noticed and criticized Stueckelberg’s paper in 1934, but only drew Heisenberg’s attention to it in 1937. See Chris Oakley’s web page on the search for a quantum field theory. He says renormalization should not be necessary, and Stueckelberg thought of it first. He comments on Peter Woit’s December 2005 blog article yet more assorted links. That’s where Woit refers to Stueckelberg and another physicist with the initials E C G: Sudarshan. And where Oakley said this: “in the case of Stueckelberg, the HEP community seems to have retained the chaff (his work on renormalization) while throwing out the wheat (his covariant perturbation theory)”. Interesting stuff.

Max Born and the foundations of a new field theory

There was more interesting stuff in 1934. Such as Max Born‘s paper on the quantum theory of the electromagnetic field. By now the scientific exodus had begun, Born was in Cambridge, and his paper was in English. He said attempts to apply the quantum theory to the electromagnetic field were open to serious objections. Like: time was treated differently to space co-ordinates and relativistic invariance was artificially imposed. Like: it’s not a self-contained theory of the electromagnetic field “but a superposition of Maxwell’s electromagnetic field on the material field of Schrödinger or Dirac, in which the elementary particles occur as point-charges”. Like: there are other difficulties such as “the infinitely great Nullpunktsenergie which is avoided by an artificial modification of the formalism”. He said another set of equations were required, which were a special form of Gustav Mie’s general field theory. On page 8 Born talked about spherical waves. On page 17 he referred to Pauli’s critical remarks on Mie’s theory. He soon followed up with a paper called foundations of the new field theory co-authored with Leopold Infeld. This was the Born-Infeld model. They started by talking about the unitarian standpoint, which assumes only one physical entity, the electromagnetic field, wherein “matter particles are considered as singularities of the field”.

GNUFDL image by AllenMcC, see Wikipedia commons and the Wikipedia vector field article

Then they talked about the dualistic standpoint which “takes field and particle as two essentially different agencies”. They said this standpoint says particles are the sources of the field, are acted on by the field but not part of the field. And that “at the present time nearly all physicists have adopted the dualistic view”. And that quantum mechanics “in its present form is essentially based on the dualistic view”. Born and Infeld also said this: “quantum electrodynamics meets considerable difficulties and is quite insufficient to explain several facts. The difficulties are chiefly connected with the fact that the self-energy of a point charge is infinite”. And this: “considerations of this sort together with the conviction of the great philosophical superiority of the unitarian idea have led to the recent attempt to construct a new electrodynamics”. Again, it’s interesting stuff.

Born and Infeld take it further

Born and Infeld’s 1934 paper on the quantization of the new field equations I is interesting too. They talk about D and B a great deal. Their 1935 paper on the quantization of the new field theory II is even more interesting. On page 12 they said this: “the inner angular momentum plays evidently a similar role to the spin in the usual theory of the electron. But it has some great advantages: it is an integral of the motion and has a real physical meaning as a property of the electromagnetic field, whereas the spin is defined as an angular momentum of an extensionless point, a rather mystical assumption”. Music to my ears. On page 17 they said this: “we think that the value of Dirac’s theory lies more in mathematical advantages than in its physical significance. It is well known that other systems with spin ½ do not obey Dirac’s laws; even the simplest one, the proton, has properties contradicting the consequences of Dirac’s equation”. And this: “the rest-mass occurring in our theory is not, as in Dirac’s, an absolute constant of the system but the total internal energy, depending on rotation and internal motion of the parts of the system. An external field will influence not only the translational motion, but also these internal motions”. On page 23 they said this: “in the classical theory we got the result S = D x B = E x H”. They’re talking about the Poynting vector. Hallelujah.

The Breit-Wheeler process

Only I don’t think the major players were listening, either to Born or others making the same sort of noises. In 1933 Otto Halpern had written a very brief paper on the scattering processes produced by electrons in negative energy states. He talked about hard gamma rays and said in the frame of reference where the two quanta have the same frequency, “the scattering process would simply consist of a rotation of the line indicating the direction of propagation through a certain angle”. In 1934 Gregory Breit and John Wheeler described gamma-gamma electron-positron pair production in their paper collision of two light quanta. The word “virtual” is not mentioned. However in 1935 Heisenberg’s students Hans Euler and Bernhard Köckel wrote a paper on the scattering of light by light in Dirac’s theory. They started by saying “there will be processes in which two light quanta create a virtual pair (positron and electron), which then immediately radiates again”. They finish by saying “we would like to cordially thank Herrn Prof. Heisenberg for the problem definition, his ongoing interest, and his numerous suggestions on the work”.

Virtual particles morph into real particles

Euler wrote a further paper in 1936 on the scattering of light by light in Dirac’s theory. It’s more of the same, but worse. There are sixteen instances of “virtual”. He said “two electrons can create light quanta and thus introduce a mutual interaction, which perhaps expresses the scattering of electrons from each other, and which gives an intuitive expression for the Coulomb law in a certain approximation”. He also said “likewise, two light quanta create a set of virtual pairs and there thus exists an interaction between them that leads to the scattering of light by light”. He also claimed that “just as in the ordinary Maxwell theory an electron is surrounded by an electromagnetic field, so is the light quantum in hole theory surrounded by a matter field”. And again he finishes up saying ”I would like to express my heartfelt thanks to Herrn Prof. Heisenberg for his essential assistance and his continuing interest in my work”. Thanks a bunch, Werner. I don’t imagine Heisenberg was paying any attention to Myron Mathisson’s 1937 paper the jittering electron and its dynamics either. I imagine he dismissed it, just as he dismissed Schrödinger’s wave theory as “mist”, which has been translated as bullshit. Even though he knew that “with regard to quantum electrodynamics”, everything was wrong.

I regard the Dirac theory as learned trash

Take a look at Abraham Pais’ 1989 essay on the Dirac theory of the electron (1930-1936). He quoted Heisenberg in 1934 saying “I regard the Dirac theory… as learned trash which no one can take seriously”. And saying this in 1935: “with regard to quantum electrodynamics, we are still at the stage in which we were in 1922 with regard to quantum mechanics. We know that everything is wrong. But in order to find the direction in which we should depart from what prevails, we must know the consequences of the prevailing formalism better than we do”. Well how about that? But then Pais said Heisenberg “gives for the first time the foundation for the quantum electrodynamics of the full Dirac-Maxwell set of equations in the way we know it today”. There’s a contradiction there. Pais also said a year and a half later Heisenberg completed his next and last paper on positron theory. It was conclusions from Dirac’s theory of the positron co-authored with Hans Euler in late 1935. There are four instances of the word “virtual” in this paper. They said “the fact that electromagnetic radiation can be transformed into matter and vice versa leads to fundamentally new features in quantum electrodynamics”. Even though Newton knew this. And they didn’t explain how this transformation is performed. It’s still magic. They ended up saying “without doubt, the theory of the positron and the present quantum electrodynamics must be considered as temporary”. Only here we are eighty-three years later. It would seem that it wasn’t.

Weisskopf gives a summary

See Weisskopf’s 1936 paper on the electrodynamics of the vacuum on the basis of the quantum theory of the electron. He talked about pair production, saying a light quantum “can by the existence of other electromagnetic fields in empty space, be absorbed and converted into matter, in the form of a pair of electrons with opposite charges”. The vibe here is that field A absorbs energy from field B, such that field A quanta are created and field B quanta are destroyed. He also talks about the vacuum and says fields can be superimposed independently in it. That’s very different to Einstein’s 1929 history of field theory where he described a field as “a state of space” and said it can “scarcely be imagined that empty space has conditions or states of two essentially different kinds”. Weisskopf also said “if high-frequency light can be absorbed into electromagnetic fields then so must one expect the scattering or deflection of a light ray whose frequency is not enough for pair creation”. He was talking about light changing direction when it interacts with light, but sadly he didn’t follow it through to gamma-gamma pair production. Instead he said “the basic assumption of Dirac’s theory of the positron is that the physical behaviour of the vacuum can be described, in a certain sense, by the behaviour of an infinite set of electrons – the vacuum electrons – that are found in states of negative energy and collectively define its state”. This is the Dirac sea again, now morphing into virtual particles. Weisskopf did say the determination can’t be complete if the vacuum electrons possess infinite charge. That’s good. But he went on to say pair production can be regarded as a jump of a vacuum electron into a state of positive energy, which isn’t. Nor is the fact that he mentioned Born and Infeld’s 1934 paper on page 16 but got the citation wrong. I don’t think he took anything on board from Born and Infeld.

Where did it all go wrong?

I don’t think Dirac did either. Even though as Schweber said on page 86 of QED and the men who made it, quantum field theories were plagued with divergences. On page 87 he talked about the point model of particles, and said the idea of looking for a structure of the electron was given up because, “as Dirac suggested (1938b), the electron is too simple a thing for the question of the laws governing its structure to arise”. Born was in Cambridge, Dirac was in Cambridge, Heisenberg was still into minimum lengths, and QED was going nowhere. In 1939 Oppenheimer’s student Sidney Dancoff wrote a paper on radiative corrections for electron scattering, but it ended with uncertainty. As Jo Bovy says in his 2006 essay the self-energy of the electron: a quintessential problem in the development of QED, physicists were overwhelmed by the problems, and believed that a conceptual change was needed to overcome them. It didn’t happen.

There’s still no concept of what an electron is

Bovy says Weisskopf gave a good review in his 1939 paper on the self-energy and the electromagnetic field of the electron, but there’s still no concept of what an electron is. In 1977 Steven Weinberg wrote the search for unity: notes for a history of quantum field theory. On page 10 he said this: “throughout the 1930s, the accepted wisdom was that quantum field theory was in fact no good, that it might be useful here and there as a stopgap, but that something radically new would have to be added in order for it to make sense”. That didn’t happen either. In more things in heaven and earth you can read an essay by Frank Wilczek. On page 158 he said there “came a long period of disillusionment”. And that initial attempts to deal with the infinities “led only to confusion and failure”. And that “quantum electrodynamics was resurrected in the late 1940s”. Which means at the end of the 1930s, it was dead and buried.

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