# User:Riccardo Guida/Proposed/Quantum field theory: origins

Over the past two decades or so, there has been a growing interest on the part of historians and philosophers of physics in the history of relativistic quantum mechanics, and more particularly, in the **history of quantum field theory** (QFT). There is now available a significant literature that addresses the developments of QFT and analyzes its conceptual foundations. The books by Pais (1986), Cushing (1990), Schweber (1994), Auyang (1995), Teller (1995), Cao (1997) and the proceedings of the conferences on the history and philosophy of high energy physics and QFT [L.M. Brown 1983, 1989, 1995; H.R. Brown and Harré 1988, Saunders and H.R. Brown 1991, Cao 1999] all attest to the fact that the subject is becoming of importance to historians and philosophers of physics as well as to physicists.

Although the history of the quantum theory -- from its beginnings in 1900 to its momentous culmination in 1925-7 -- has been the subject of intense scrutiny [see for example, Mehra and Rechenberg volumes 1-5, 1982-1996 and the references therein and more recently Joas et al 2008], we know far less about the genesis of QFT. For the most part, the general accounts have been written by the participants themselves. [See for example Heisenberg 1930, Pauli 1933, Wentzel 1960 and Pais 1986]. We do have careful historical accounts of Dirac's formulation of QED in 1927 and of his relativistic wave equation for a spin 2 particle and its reinterpretation as *hole theory* [see Kragh (1990) and the references therein, Dalitz 1995} but there is no thorough historical and philosophical analysis of the developments of QFT beginning with Heisenberg and Pauli's (1929, 1930) and Fermi's (1929,1930, 1932) formulations of QED that covers the development of the field during the 1930s. Olivier Darrigol made a start in such an enterprise in an impressive thesis written in 1982, which covered the period from 1927 till the early 1950s, but the material was not converted into a book [see however Darrigol (1984, 1984, 1988)]. In 1958 Schwinger edited a very useful source book on quantum electrodynamics (QED) for which he provided an informative introductory essay covering some of the developments during the thirties. But he was primarily concerned with charting the path to the post World War II developments and illustrating the effectiveness of the renormalization procedures and the power of the new calculational tools that Feynman, Dyson and he had produced. Similarly, Miller (1994) has edited a valuable source book containing many of the important papers published during the 1930s dealing with the divergences encountered in QED, but like Schwinger's source book the selections were chosen with the post World War II developments in mind. More recently Pais (1986) gave a very informative, technical account in Inward Bound, but shunned philosophical issues. Mehra and Rechenberg in Volume 6 of their history of the quantum theory have given an overview of the developments in QFT from its inception till the outbreak of World War II. ( Mehra and Rechenberg vol.6, 2000), but their account will be the point of departure for future researches as their presentation is detailed rather than synthetic. Mention should also be made of Laurie Brown and Helmut Rechenberg's important, detailed studies of the quantum field theoretic description of nuclear forces during the decade following Yukawa's initial paper (Brown, L.M. 1981, 1985, 1991a, b). A comprehensive version of their researches appeared in book form in 1996 and is a most valuable contribution to our understanding of the history of QFT during the 1930s.

The following is a brief sketch of some of the developments in QFT during the 1930s which emphasizes aspects of the history relevant to renormalization theory, and which at various points raises certain historiographical questions.

## Contents |

## Some aspects of the history of QFT during the 1930s

Two dozen or so people made up the cast of actors as the story unfolds until the mid 30s. Of those concerned with foundational issues Dirac, Heisenberg, and Pauli played central roles, with a dozen or so of others -- notably Bethe, Bohr, Born, Breit, Rosenfeld, Fermi, Heitler, Jordan, Klein, Møller, Oppenheimer, Peierls, Fock, Tamm, Rosenfeld, Waller, Weisskopf, Wentzel, and Wigner -- making important contributions.

With the elucidation of the mechanism of cosmic ray showers which led to the discovery of the *mesotron* [ the particle we now call the \(\mu\)-meson [see Cassidy (1981),L.M. Brown and Hoddeson (1983), Galison (1987)], and its identification with the particle associated with the Yukawa's scalar field responsible for nuclear forces, the field took on a different character [see especially L.M. Brown and Rechenberg 1996, chapter 6]. From 1936/7 on, it is principally the problem of the nuclear forces and associated mesons that appropriated center stage. The realization that the *building blocks* of the world included more than protons and neutrons, electrons and positrons, photons and neutrinos marked the beginnings of high energy and elementary particle physics. Although QED retained a privileged position as the paradigmatic quantum field theory, after the discovery of a mesotron in the cosmic radiation, and the belief that it corresponded to the quantum that Yukawa had postulated to account for the nuclear forces, it became clear to Heisenberg, Oppenheimer, and most other theorists that the important foundational problems of theoretical physics were now concerned with meson theories. Empirical data referring to length scales of the order of 10-13 cm coming from experiments analyzing cosmic rays, \(\beta\)-decay, proton-proton and neutron-proton scattering, became the foci of the theorists' attention. And in addition to the theorists mentioned earlier, the theorists contributing to the developments now include Yukawa, Sakata, Tanikawa, Tomonaga, ... in Japan; Fröhlich, Heitler, Kemmer, in Great Britain; Fierz, Stückelberg in Switzerland; and a sizable number of younger American theorists, among them, Critchfield, Kusaka, Dancoff, Morrison, Schwinger,...

Miller's source book gives a good indication of the principal QED foundational problems addressed during the 1930s. Except for Heisenberg's 1938 paper in which he speculated about the possibility of a fundamental length^{1} all the articles Miller has reprinted deal with the foundational aspects of the quantum mechanical description of the interaction of electrons and positrons with the electromagnetic field. Furthermore, since except for Kramers's 1938 paper, all the articles reprinted in Miller's source book are either by Dirac, by Heisenberg, or by Pauli or by Pauli and/or his assistants -- namely, Weisskopf and Fierz -- we do not obtain an adequate picture of the contributions of other workers in the field such as Tamm, Akhiezer, Serber,.. . If any one person dominates Miller's account it is Heisenberg. Gauged by the importance of his contributions and by the heroic efforts he expended on QED until the mid 1930s this is surely justified. It also allowed Miller to draw heavily on the Pauli correspondence of the 1930s which can be found in volume 2 of his Wissenschaftlicher Briefwechsel that von Meyenn has so splendidly edited. In view of the important role that Heisenberg played in the developments of QFT reference should also be made to Heisenberg's Collected Works (1989) and in particular to Volume IIA which contains important unpublished materials by Heisenberg from the 1930s as well as valuable and historically sensitive assessments of his work by Bagge, Oehme, Pais, Rechenberg and others.

The articles that are reprinted in Miller's source book mark the high ground taken during the 1930s in the march towards renormalization theory. Papers 1 and 2 (by Heisenberg), 5 (by Weisskopf) and 11 (by Kramers), clarified the electron self energy problem, and the one by Kramers laid the foundations for the procedure and meaning of mass renormalization; papers 3 and 4 (by Dirac), 6 ( by Heisenberg) and 8 (by Weisskopf) indicated the novel features in hole theory connected with the polarization of the vacuum by an external field, elucidated the difficulties and intricacies encountered in that problem^{2}, and formulated a subtraction procedure for handling the divergences. Paper 9 ( by Pauli and Fierz) gave an *anschaulich* solution to the infrared problem (and coincidentally shed light on how mass renormalization is to be carried out in a non-relativistic model of a single charged particle interacting with the quantized electromagnetic field).

Two other papers are included in Miller's source book. Paper 7 is a translation of Pauli and Weisskopf's 1934 article on the quantization of the charged scalar field which constituted an important contribution to the quantum theory of wave fields. Pauli called the Pauli-Weisskopf theory the Anti-Dirac theory because of his aversion to hole theory and the *subtraction physics* it entailed. The comparison of the structure and properties of the observables of a relativistic charged spin zero field (when quantized with commutators) and those of a relativistic charged spin one-half field (when quantized with anti-commutation rules) became the basis of Pauli's famous spin-statistics theorem. In its first formulation [Pauli 1936] the theorem stated that it was possible to exclude mathematically field theories describing particles having spin zero but obeying the Pauli exclusion principle on the assumption that

- the theory is relativistically invariant;
- the operators representing physical quantities, such as the energy momentum and the charge and current density, are hermitian;
- that the commutator of the charge density operators, \(\rho(\boldsymbol{x},t)\) and \(\rho(\boldsymbol{x'},t)\ ,\) at two points separated by a space like distance vanishes [Pauli 1935].

The other paper included in Miller's source book (paper 10) is the article Heisenberg wrote for the 1938 *Festschrift* celebrating Max Planck's eightieth birthday. Its title is "The universal length appearing in the theory of elementary particles". Here mention should also be made of Heisenberg's report to the 1939 Solvay Congress (which was never held because of the outbreak of World War II). In this report, which is reprinted in Heisenberg's Collected Works Volume II, as well as in an earlier paper on cosmic ray showers published in 1936, Heisenberg pointed out the characteristic differences in the behavior of cross-sections at high energy of quantum electrodynamic and beta-decay processes by virtue of the dimension of their coupling constant. The distinction between quantum field theories with dimensional and nondimensional coupling constants was the basis of Dyson's 1949 criterion for theories to be renormalizable or non-renormalizable. [See also Sakata 1952.]

If the articles in Miller's source book represent the high ground, Fermi's papers from 1929 to 1932 represent the pragmatic, practical approach to problems in QED. In a series of papers from 1929 to 1932 Fermi formulated a relativistically invariant description of the interaction between charged particles and the electromagnetic field which treated both the particles and the em field quantum mechanically. He first devised a simple, readily interpretable, Hamiltonian description of charged particles interacting with the e.m. field, then indicated how to quantize this formulation and thereafter showed how to exploit perturbation theory to describe quantum electrodynamic phenomena.

Heisenberg and Pauli's 1929 and 1930 papers tackled these same problems, but their formulations did not have the simplicity and Anschaulichkeit, yet thoroughness of Fermi's approach. What Fermi accomplished with his papers was the following:

- He provided a simple, vizualizable way to describe the interactions between photons and charged particles. It was a formulation from which an entire generation learned on how to think about quantum electrodynamic effects in atomic phenomena. It was the point of departure in 1939 for Feynman addressing quantum electrodynamics in his Lagrangian formulation (Feynman 1941), and the source of Weisskopf's insight that the Lamb shift could be interpreted as the effect of the zero point energy vacuum fluctuations of the electromagnetic field on the motion of the electron in the hydrogen atom (Welton 1948, Weisskopf 1949). In addition, Fermi showed how to handle the gauge conditions in QED when the electromagnetic field is described by vector and scalar potentials -- and his treatment was the point of departure for fixing gauges in quantized gauged theories.
- He indicated under what circumstances the intuitive picture of a photon as a massless, spin 1 particle-like entity that moves with velocity c was appropriate. And
- in a paper with Bethe he helped secure the perturbation theoretic picture that depicts the interaction between charged particles as stemming from the exchange of photons( Bethe and Fermi 1932).

This last paper was particularly important for it formulated what is one component of the underlying metaphysics of particle physics when attempting to *explain* the phenomena by quantum field theory: the interactions between *elementary* particles stem from the exchange of quanta of the fields with which they interact^{3}. It implied that the potentials introduced to explain the Coulomb and magnetic interactions between charged particles had to be accounted for field theoretically in terms of exchanges of longitudinal and transverse photons. Similarly later in the decade, the interactions between nucleons in terms of phenomenological potentials had to be justified field theoretically in terms of exchanges of the quanta of appropriate meson fields.

Fermi's QED studies were important, if not necessary, preparations for him being able to formulate in 1933 his theory of \(\beta\)-decay and that in the first of these papers he could assert that electrons do not exist as such in nuclei before \(\beta\)-emission occurs

but that they, so to say, acquire their existence at the very moment when they are emitted; in the same manner as a quantum of light, emitted by an atom in a quantum jump, can in no way be considered as pre existing in the atom prior to the emission process. In this theory, then, the total number of the electrons and of the neutrinos (like the total number of light quanta in the theory of radiation) will not necessarily be constant, since there might be processes of creation or destruction of these light particles. [Fermi 1934]

It is interesting to note that shortly after reading Fermi's \(\beta\)-decay theory paper in the *Zeitschrift für Physik* Heisenberg wrote Pauli:

If the Fermi matrix element for the creation of an electron + neutrino pair is correct, then -- similar the case of the atomic electrons where the exchange of light quanta results in the Coulomb force -- it should give rise in second order to a force between neutron and proton. (Pauli 1985, p. 250)^{4}.

Arguably, Fermi's paper on \(\beta\)-decay constitutes the birth of quantum field theory as applied to what is now called *elementary particle physics*. It was the first *quantum field theory* formulated after QED -- and it was modeled upon QED. It emphasizes two aspects of QFT that will be crucial in all subsequent developments:

- the creation and annihilation of particles. This is explicit in Fermi's \(\beta\)-decay theory. In the case of QED, the explanation of the Meitner-Hupfield effect (Brown, L.M. and Moyer 1984) in the absorption of ThC" gamma rays in terms of electron-positron pair production was an important landmark, which indicated the novel features of relativistic QED.
- QED, whether in its hole theoretic or field theoretic formulation, and Fermi's \(\beta\)-decay theory constitute descriptions of quantum systems with an infinite number of degrees of freedom.

The pragmatic approach that permeated Fermi's presentation of QED in his 1932 *Reviews of Modern Physics* article dominated those aspects of QED that were of immediate relevance to experimental data during the 1930s. Thus Heitler's approach to perturbation theory in his *The Quantum Theory of Radiation* (Heitler 1936) and his conceptualization of electrodynamic process is based and modeled on Fermi's 1932 *Reviews of Modern Physics* article.

One of the characteristic features of the history of QFT during the 1930s must be borne in mind: throughout the decade most of the workers in the field doubted its correctness in view of the divergence difficulties that plagued all formulations of relativistic quantum field theories [Rueger 1992]. They were always searching for the correct *future theory*. In relating the story of the developments of QFT during the 1930s one of the tasks of the historian is therefore to elucidate and interpret the assumptions and hopes by which theoretical physicists were charting the course towards the future definitive theory -- and not allow the account to be guided retrospectively by an endpoint such as the formulation of renormalization techniques in the late 1940s. As an illustration of how this can be done, Cathryn Carson in her PhD dissertation has carefully and thoroughly analyzed Heisenberg's scientific work during the thirties -- encompassing all the factors that directed him to enunciate his views on the existence of a fundamental length and led him to his S-matrix papers -- and has done so without making it a Whiggish account. [Carson 1995; see also Cassidy 1992].

Although my focus is on the foundational aspects that eventually culminated in the renormalization program of the 1947-1950 period, the key role played by experiments in the development of QED ^{5} should not be overlooked (e.g measurements of the cross-sections for Compton scattering, electron-electron scattering, Bremsstrahlung, pair production, electron-positron scattering, the precise determination of the spectrum of hydrogen, ... ). These phenomena were closely related to the novel conceptualization inherent in relativistic quantum field theories and the drive for quantitative explanation of the experimental data was responsible for important advances and clarifications ^{6}. Thus, the analysis of electron-electron scattering experiments by Møller and others led Bethe and Fermi (1932) to formulate their fully quantum field theoretical, perturbation theoretic derivation of the electron-electron potential, in contrast to Møller who still used a correspondence principle argument in the description of the interaction of charged particles with the electromagnetic field. [See the articles by Kragh (1991) and Roqué (1991)]. Similarly, the important paper by Uehling (1935) on vacuum polarization stemmed from an attempt to explain the experimental findings by Houston and Hsieh (1934) on the spectrum of atomic hydrogen -- data that seemed to disagree with the prediction of the Dirac equation. Pauli in 1935, in a presentation describing the Pauli-Weisskopf theory, the QFT of charged scalar particles, in the seminar on "The Theory of the Positron and Related Topics" that he had organized at the Institute for Advanced Study in Princeton during 1935-6, noted that

if we want a quantitative theory of pair production [ of charged scalar particles] we must, as in the theory of holes, introduce second quantization. In the theory of holes this step is necessary in order to make transmutation between energy and matter discrete instead of continuous. The introduction of second quantization in the scalar theory makes no difference to the annihilation of matter, but it is necessary for a good theory of the production of pairs. This is quite analogous to the effect of the field quantization on the theory of absorption and emission of radiation." Pauli in Hoffman (1936), p. 4.

During the thirties the quantum theory of fields was seen to run into difficulties because one was dealing with systems with an infinite number of degrees of freedom. Yet it is not quite clear how this transition was conceptualized: Was it thought that the mathematics developed for the rigorous formulation of non-relativistic quantum mechanics is adequate for describing field systems? Various people, and in particular Pauli and von Neumann, raised the question whether the canonical Hilbert space formalism of the quantum theory was adequate for describing field systems. Thus Pauli in 1933 in the final section of his Handbuch article after discussing the self energy problem asserted that "the use of the wave-mechanical formalism to a system with infinitely many degrees of freedom violates the correspondence principle. Herein we may see a hint that not only the field concept but also the space time concept in small regions may require a fundamental modification." (Emphasis in the original). Similarly, von Neumann came to believe that "the use of wave functions, or in the language of Dirac, of maximum observations" was wrong when dealing with systems with an infinite number of degrees of freedom: one would need an infinite number of compatible measurements to determine its state. von Neumann went on to develop a model that avoided wave functions. In the process he developed the mathematics of infinite direct product of Hilbert spaces. [See von Neumann 1932, 1936, 1938. Also his lecture in Hoffman 1936]. A related question that was raised early in the decade was the following: "Can it be presumed that the quantum theory of measurement as dictated by Bohr -- with its requirement of classically describable macroscopic measuring devices -- is adequate for quantized field systems?" Some answers to that question were given by Landau and Peierls [Landau 1931, see also Pauli 1980, p.] and will be referred to in the next section. Later in this essay I shall try to answer the question: "What was the description and the conceptualization of a charged particle (such as an electron or a positron) in (early) QED?"

## Divergences

Already in the first Dreimannerarbeit Pascual Jordan had investigated the quantization of a system with an infinite number of degrees of freedom, namely a vibrating string with fixed ends. Classically the string is equivalent to an infinite set of (harmonic) oscillators (described by canonical variables \(q_k\ ,\) \(p_k\)). In the quantum theory ^{7} these are treated as quantized oscillators, which in turn implies if the Hamiltonian of an oscillator is \(1/2(p_k^2 + q_k^2\)) its ground state has a *zero-point* energy of \(1/2\omega_k\ .\) For the quantized string this led to the first divergence difficulty of the quantum theory of fields: the zero point energy of the ground state is infinite since there are an infinite number of oscillators. Jordan dropped this infinite constant and this constituted the first infinite subtraction in the history of *subtraction physics*, the precursor during the thirties of the post-war renormalization theory. The justification for the dropping of the term was that the procedure yielded finite answers for the difference of the energies of two different states of the system.

Jordan and Pauli (1928) carried out this same procedure for the free electromagnetic field, and there too encountered the infinite zero point energy and dropped it. They also found that the quantization procedure led to certain commutation rules for the field strengths. The deletion of the zero point energy in the case of a single harmonic oscillator does not mean that there are no zero point vibrations. These imply that measurements of the observable \(q(t)\) in identically prepared systems of a (quantized) harmonic oscillator will give different results. Similarly, in the case of the quantized electromagnetic field the measurement of \(E(\boldsymbol{x},t)\ ,\) the electric field intensity at a given point, will tend to differ for identically prepared electromagnetic field systems. The electric field has quantum fluctuations. In fact, for the quantized electromagnetic field the mean square value of the electric (or magnetic) field at a given space time point in the ground state (i.e. for the vacuum), diverges. For quantized fields only quantities like \(\int E(\boldsymbol{x},t)f(\boldsymbol{x},t)dx dt\) (where \(f\) is a smooth function equal to \(1\) during the time interval \((t,t+T)\) inside some region \(R\) around the point \(x\) and converging rapidly to zero outside of \(R\)) make sense as operators on states. This in fact was the central idea in Bohr and Rosenfeld's famous paper on the measurability of the electromagnetic field. We do not have as yet a thorough account of this oft quoted but seldom read classic article. It is important to have a full analysis and understanding of Bohr and Rosenfeld's paper because measurability bears on what is considered real and their analysis is thus central to clarifying what is meant by the quantum em field (and by extrapolation what is meant by a quantized field.) For the (free) em field their phenomenological point of view required non-atomistic, classically describable, test bodies whose dimensions are large compared to their Compton wave length, and whose charge is large compared to \((hc)^{1/2}\) (i.e. large compared to \((137)^{1/2} \times\) the electronic charge). Under those circumstances Bohr and Rosenfeld showed that it is possible to consistently carry out measurements of field strengths in regions comparable to the size of the test body. But this is of course a highly idealized setup, with the success of the analysis stemming from the smallness of the fine structure constant, \(e^2/hc = 1/137\ ,\) and from the fact that the quantum theory of the free electromagnetic field contains no characteristic length or mass: the theory is independent of any limitation or abstraction of matter.

Darrigol (1991) ^{8} has given the historical background of the genesis of Bohr and Rosenfeld's paper stemming from the challenges posed by Jordan and Fock (1930) and by Landau and Peierls (1931), and has indicated how Bohr's theoretical investigations of the measurability of electric and magnetic field intensities were related to his conception of the consistency and completeness of the quantum theory of the free electromagnetic field. Similar considerations were given to the measurement of the charge-current and energy densities, entailing the calculation of the fluctuations of the charge and the energy in small regions of space-time ^{9}. What is needed is a full account of the various fluctuation phenomena that were investigated during the early days of QED, with an explanation of what they meant and a clear description of how the divergences encountered in these investigations were interpreted and tamed. The analysis of the energy of the vacuum state and of the vacuum fluctuations in QED gave rise to an *anschaulich* account of quantum electrodynamic physical effects ^{10} and culminated in the papers of Casimir (1948), Welton (1948) and Weisskopf (1949) ^{11}. It would be good to have the history of this viewpoint.

We also need to better understand the physical, mathematical and philosophical justification theorists gave for their particular methods for eliminating infinite quantities. Changing the concepts of space and time was one possibility. Heisenberg's introduction of a fundamental length (see papers 1 and 11 in Miller's source book) was to demarcate the scale at which such modifications were to come into play ^{12}. Similarly, it would be good to have a thorough account of the *subtraction physics* developed to deal with the divergences stemming from vacuum polarization and clarify its relation to the post World War II renormalization program.

Incidentally, renormalization has come to mean a variety of things and often its usage includes particular procedures by which infinite or indefinite terms in QFT are replaced by well defined mathematical objects. But one should differentiate between *regularization* -- algorithms advanced to produce finite answers -- and *renormalization*. Renormalization consists of a set of rules that produce finite answers, together with the physical grounds for the procedures, justifications that are related to the consistency and other aspects of the theory. In its broadest sense renormalization theory subsumes all the various methods of analyzing the parameters labeling the states of a theory and of the relation of these parameters to the parameters appearing in the Lagrangian defining the theory and to experimentally determined data [Wightman 1985]. It would be valuable to have a concise history and analysis of the various regularization procedures advanced during the thirties. Separating the regularization considerations from the renormalization aspects of the "subtraction" physics would be helpful when assessing the post war developments. It would undoubtedly result in a more continuous account of the evolution of QFT from the 30s to the late 40s ^{13}.

Articles published in the *Zeitschrift für Physik, Physical Review, Proceedings of the Royal Society* , ... are not necessarily the most transparent sources for unraveling and understanding the history of the developments. The private correspondences, working notes, drafts of papers, proceedings of conferences, ... are valuable sources for such historical undertakings. We will make available such documents on our web site. An example of such a document are the reports of the presentations in the seminar that Pauli ran at the Institute for Advanced Study during the academic year 1935-36. The notes were edited by Banesh Hoffman and issued as "The Theory of the Positron and Related Topics". They constitute a first hand account of the concerns of the people working in QED in 1935. These notes make clear that besides the problems of the self-energy of the electron and of vacuum polarization connected with hole theory, the neutrino theory of light, pair production by photons in the field of a nucleus, the limits of applicability of QED, Born-Infeld theory and its quantization, the quantum mechanics of infinite systems were problems on which considerable efforts were expended. The informal nature of the presentations give insights that are difficult to obtain from refereed articles. I next turn to some of the specific problems addressed in QED.

### The self energy of the electron and of the photon

In the introductory statement in his paper on the self-energy of the electron Heisenberg commented that:

In classical theory, the field strengths \(E\) and \(H\) become arbitrarily large in the neighborhood of the point-charge e, so that the integral over the energy density \((1/8\pi)(E^2 + H^2)\) diverges. To overcome this difficulty, one therefore assumes a finite radius \(r_0\) for the electron in classical electron theory. The radius is related to the mass \(m\) of the electron in the order of magnitude relation \(r_0 - e^2/mc^2\ :\) the integral over the energy density is then of the order \(mc^2\ .\) In quantum theory, not only this radius \(r_0\) but possibly another length \(\lambda_0 = h/mc\ ,\) which is characteristic of the electron, plays a role in the self-energy. In a superficial consideration in terms of the correspondence principle, one would suspect that the self energy of the point-like electron must also become infinite in quantum theory. In fact Oppenheimer (1930) and Waller (1930) have indeed shown that a perturbation method which proceeds in powers of \(e\) does not yield finite values for the self-energy [of a point-like electron].

But instead of considering a finite sized electron, Heisenberg suggested

that one divide space into cells of the finite size \(r_0\ ,\) and that one replaces the present differential equations by difference equations. ..[The] self energy of an electron would be finite in such as lattice world. [But] the statement that a smallest length exists is no longer relativistically invariant, and no way is presently known to harmonize the requirement of relativistic invariance with the fundamental introduction of a smallest length. In the meantime it therefore seems more correct not to introduce the length \(r_0\) into the foundation of the theory but to hold fast to relativistic invariance.

Heisenberg then proceeded to analyze the self-energy of an electron moving with a speed nearly that of light, in which case its rest mass can be neglected, and thus "we always calculate ... with \(m = 0\ .\)" Under those circumstances Heisenberg believed that the self energy must remain finite on dimensional ground.

The one particle Hamiltonian Heisenberg worked with is given by:

\[ \begin{array}{lcl} H = \mathbf{\alpha} \cdot (\mathbf{p} + \frac{e}{hc}\mathbf{\Phi}) + \int dV \frac{1}{8\pi} (\mathbf{E}^2 + \mathbf{H}^2)\\ curl \Phi = \mathbf{H}\\ div \mathbf{E}=4\pi e \delta (\mathbf{x}-\mathbf{q}) \end{array} \]

with \(\mathbf{p}\) and \(\mathbf{q}\) and \(\mathbf{\Phi}\) and \(\mathbf{E}\) satisfying the usual commutation rules. The total momentum operator for the system is given by

\[ \mathbf{G} = (\mathbf{p} + \frac{e}{hc}\mathbf{\Phi}) + \int dV 1/2 [\mathbf{E}\times\mathbf{H} - \mathbf{H}\times\mathbf{E}] \]

Heisenberg then noted that the electron coordinates can be completely eliminated from the Hamiltonian by using the total momentum so that it becomes:

\[ \mathbf{H}=c\alpha\cdot\mathbf{G} - \alpha \cdot (\frac{1}{4\pi} \int dV 1/2 [\mathbf{E}\times\mathbf{H} - \mathbf{H}\times\mathbf{E}]) + \frac{1}{8\pi} \int dV (\mathbf{E}^2 + \mathbf{H}^2) \]

For an electron under the influence of no force Heisenberg then claimed that the equation

\[ H = c\mathbf{\alpha}\cdot\mathbf{G} \]

must hold. He therefore looked whether solutions satisfying

\[ [\int dV \mathbf{\alpha} \cdot [\mathbf{E}\times\mathbf{H} - \mathbf{H}\times\mathbf{E}] + \int dV (\mathbf{E}^2 + \mathbf{H}^2)]\mathbf{\Psi} = 0 \]

exist -- and came to the conclusion that

[the] one electron problem could be treated correctly this way without an infinite self energy if there were solutions of vacuum electrodynamics without a zero-point energy. Unfortunately, such solutions do not exist....A solution of the basic equations ...has therefore not been found for the time being; it is also not probable that one will achieve a solution without substantial modification of the quantum theory of wave fields.The purpose of this paper was to show that the difficulties of field theory do not come directly from the infinite self-energy of the electron but that rather the foundations of field theory still require modification. (emphasis added)

In a remarkable paper written slightly later, Fermi (1931) also investigated the problem of the self-energy of an electron. Fermi was aware that the divergences resulted from the point character of the charges which was reflected in the local nature of the stipulated interaction\[\mathbf{p}_i.\mathbf{A}(\mathbf{x}_i)\] or \(\mathbf{\gamma}_i.\mathbf{A}(\mathbf{x}_i)\) and in the form of the Coulomb interaction. Fermi undertook to explore the consequences of assuming that the charge on an *elementary* particle was extended -- fully aware that this destroyed the relativistic invariance of the theory. His model is similar to Lorentz's: electrons were entities with finite extension. Moreover, Lorentz (1916) made a distinction between electromagnetic mass, \(m_{ele}\ ,\) and material mass, i.e. mechanical mass, \(m_0\ ,\) the electromagnetic mass being the inertia that the charged particle had by virtue of its charge. The total mass, identified with the experimentally determined mass, is

\[ m_{exp} = m_0 + m_{ele} \]

Lorentz thought that all the mass of the electron was electromagnetic, i.e. that \(m_0 =0\ .\) Fermi in his investigation echoed these views. Fermi made the charge frequency (scale) dependent to reflect its distributed nature. His argument for doing so was as follows: if the electron has a finite radius its various parts will present the same phase as far as the wave lengths that are large in the comparison with the electron radius are concerned. On the other hand, for wave lengths of the order of or smaller than the electron's radius different interior points will react with different phases. The electron thus interacts differently with radiation of different frequency -- effectively it presents a smaller charge for high frequencies -- with the observed charge some kind of average charge. Thus in the Hamiltonian Fermi makes the charge of each particle frequency dependent \[e_i = e_i(\nu)\ .\]

Following Heisenberg, Fermi when characterizing the one particle state as having a charge \(e\ ,\) momentum \(\mathbf{p}\) and energy \(E\ ,\) required that this state vector must be an eigenfunction of the *total momentum* operator, \(\mathbf{G}\ ,\)(i.e. of field + of particle + momentum stemming from the interaction between charge and field ) with eigenvalue \(\mathbf{p}\ .\) He then investigated the chiral case, [i.e. the formulation when the mechanical mass in the Dirac equation for the charged particle is set equal to zero], to see whether he could find states \(\mathbf{\Phi}\) which satisfied the requirements

\[ \mathbf{G}\mathbf{\Phi} = \mathbf{p}\mathbf{\Phi} \]

and

\[ H\mathbf{\Phi} = E\mathbf{\Phi} \]

The joint requirement can be satisfied since \(\mathbf{G}\) and \(H\) commute with one another \[[\mathbf{G} , H ] = 0\ .\]). In fact, Fermi explicitly exhibited the state vectors that satisfied the equation \(\mathbf{G}\mathbf{\Phi} = \mathbf{p}\mathbf{\Phi}\ .\) He then pointed out where Heisenberg had gone wrong. Heisenberg's assumption that in the full quantum electrodynamical description the force- free electron must satisfy the Dirac equation

\[ H = c\alpha \cdot \mathbf{G} \]

was incorrect. As a result of its interaction with the [quantized] electromagnetic field the equation the electron obeys should be taken to be

\[ H = c\alpha \cdot \mathbf{G} + \beta m_{em} \]

where \(m_{em}\) is the electromagnetic mass the electron acquires by virtue of the interaction. Fermi therefore inquired whether there are solutions that satisfy

\[ H\mathbf{\Phi} = c \sqrt{\mathbf{p}^2 + m^2_{em} c^2 \mathbf{\Phi}} \]

with \(m_{em}\) to be determined. Fermi formulated a perturbation theoretic approach, and found that to lowest order an approximate solution exists for which

\[ m_{em} = \sqrt{\frac{4\hbar}{c^2} \int_0^\infin e^2(\mathbf{\nu}) \nu d\nu)} \]

Fermi commented that by virtue of the dependence on \(h\) the generation of the electromagnetic mass was a strictly quantum mechanical phenomenon. Note that the chiral symmetry is broken! Fermi thus discovered an early case of anomalous or quantal symmetry breaking, namely that the symmetry of the classical theory need not survive quantization. The introduction of the frequency dependent charge was a regularization procedure that allowed the symmetry breaking to be exhibited.

As far as I have been able to ascertain, Fermi's paper fell on deaf ears. During the 1930s -- except for Kramers' and Pauli and Fierz' papers in 1938 -- relativistic invariance took precedence over structural modeling and calculations. Thus there is no reference to either Heisenberg's 1930 self-energy paper nor to Fermi's 1931 paper in Weisskopf 1934 papers wherein he calculated the self-energy of the electron in hole theory and ascertained that to order \(\alpha = e^2/hc\) the self-energy diverges logarithmically ^{14}. The Weisskopf calculation, in its corrected version due to the Furry contribution, was a landmark in QED during the 1930s. Incidentally, in a subsequent paper Weisskopf (1939) suggested that the logarithmic divergence of the electron's self energy was true to all order of perturbation theory.

The disappearance of extended Lorentz-like models and the move to focusing on questions regarding *observable* quantities, such as \(m_{exp}\ ,\) extended the philosophical tradition that had motivated Heisenberg in his formulation of matrix mechanics to QFT. I shall return to this issue in our discussion of periodization.

Dirac in 1938 explicitly advised giving up the idea of looking for a structure of the electron because

...the electron is too simple a thing for the question of the laws governing its structure to arise, and thus quantum mechanics should not be needed for the solution of the difficulty [that arise in view of its assumed pointlike structure].

Dirac's statement can be interpreted as saying that if one wants to remain in the domain where a classical, non-quantum mechanical description is valid, one cannot include those features of the electron that manifest themselves at very small distances where its structure is relevant: in a classical theory the electron should be considered point like and described as a point charge. [For a review of Dirac's researches in elaborating these ideas see Rohrlich 1966, and 1973]. The issue of bypassing questions relating to the structure of the electron was explicitly addressed in Kramers' address to the Galvani Congress (Kramers 1938):

I have tried to present the theory [of the interaction between the radiation field and charged particles ] in such a fashion that the question of the structure and the finite extension of the particles are not explicitly involved and the quantity that is introduced as the 'particle mass' is from the beginning the experimental mass.

This will turn out to have been the key insight in how to handle the self-energy divergences in QED and is the foundation for what later will be called mass renormalization. Kramers had been concerned with this problem ever since the advent of quantum mechanics and he had been critical of Dirac's 1927 formulation of QED. He devoted some 80 pages of the second volume of his textbook on quantum mechanics (Kramers 1938) to what he considered to be the proper formulation of the interaction between electrons and the electromagnetic field. This formulation is explained and analyzed in Dresden's biography of Kramers (Dresden 1987), where the chronology of Kramers' obsession with this formulation is also given.

Note that Kramers' and Dirac's program share a common view: One must first obtain a consistent classical theory of the interaction of charged particles with radiation in which only observable quantities enter the formulation -- and once this was accomplished the theory would then be quantized.

## Vacuum polarization

Heisenberg thought that Dirac's hole theory with its implication of the existence of antimatter was one of the biggest revolutions of the big revolutions in physics of the 20th century. It was a discovery that changed the conception of matter. Hole theory implied that a light quantum could create an electron positron pair, which formally meant that the number of particles with a given charge was no longer a good quantum number and therefore that there was no longer a conservation law for the number of particles. Another consequence of Dirac's hole theory was that the vacuum became a complex entity: it was a state filled with an infinite number of particles of negative energy that were not observable. In his interview with Kuhn Heisenberg spoke of the mood of the early thirties:

It is very difficult to describe that state because it was psychologically so different from the state in 1923 or 1924. In 23 and 24 we knew there were difficulties and we also had the feeling that we were quite close to the final solution of these difficulties. Just one step and we will be in the new field. It was as if we were just before entering the harbor, while in this later period we were just going out into the sea again, i.e. all kinds of difficulty coming up and really we didn't know where it would lead to. And even if new and good ideas came up, these ideas would work a short way and then again one had new difficulties. It was clearly seen that this was now an entirely new story. So nobody expected quick results at that time." [Heisenberg 1963]

Once the problem of transitions to negative energy states had been solved by Dirac's *hole theory* the next difficulty proved to be the problem of the polarization of the vacuum. Dirac presented a paper on the "Theory of the Positron" at the seventh Solvay Congress that took place in Brussels from the 22nd to the 29th of October 1933. In his opening remarks Dirac characteristically noted that "The present quantum mechanics cannot be expected to apply to phenomena in which distances of the order of the classical radius of the electron, \(e^2/mc^2\ ,\) are important, since the present theory cannot give any account of the structure of the electron..." Dirac then reviewed his formulation of the hole theory, sharpening his previous formulation somewhat so that it could readily be applied to the problem at hand. "A hole can be described by a Schrödinger wave function like that describing the motion of an electron, and second, that the hole is deflected by a field in the same way as positive energy electron." Dirac also stated two important inferences: "The mass of the positron must exactly equal that of the electron and its charge must be exactly opposite to that of the electron." A further assumption Dirac made was that the distribution of electrons in which all positive energy states were unoccupied and all negative energy states occupied produce no electric field. Only *departures* from this distribution produced a

\[ div \mathbf{E} = -4 \pi (\rho - <\rho>_{vac}) \]

field in accordance with Maxwell's equation.

The question Dirac then addressed was: What happens when there is an external electric field present? In a region of space in which there is no field the division of the states into positive and negative energy states is unambiguous and the above stipulations do not result in any difficulties. However, "when applied to space in which there is an electromagnetic field,... one must specify just which distribution of electrons is assumed to produce no field and one must also give some rule for subtracting this distribution from the actually occurring distribution in any particular problem" to get a finite difference to be used in the right hand side of the \(\mathbf{L}.\mathbf{E}\) equation "since in general the mathematical process of subtracting one infinity from another is ambiguous." Dirac went on to consider the case of a weak, time independent electrostatic field, “in which the necessary assumptions seem fairly obvious." The field was assumed to be sufficiently weak "for a perturbation method to be applicable." He then found that the distribution which produces no field does not satisfy the equations of motion. By subtracting this distribution from that distribution which does satisfy the equations of motion and which corresponds to a state with no electrons or positrons present Dirac was left with "a difference which can be interpreted physically as an effect of polarization of the distribution of negative energy electron by the electric field."

The physical meaning of these results was summarized in a letter Dirac wrote Bohr after he completed of his Solvay report:

Dear Bohr, Peierls and I have been looking into the question of the change in the distribution of negative energy electrons produced by a static electric field. We find that this changed distribution causes a partial neutralization of the charge producing the field. If it is assumed that the relativistic wave equation is exact, for all energies of the electron, then the neutralisation would be complete and electric charges would never be observable. A more reasonable assumption to make is that the relativistic wave equation fails for energies of the order \(137 m c^2\ .\) If we neglect altogether the disturbance that the field produces in negative energy electrons with energies less than \(137 m c^2\ ,\) then the neutralization of charge produced by the other negative energy electrons is small and of the order \(1/137\ .\) We then have a picture in which all the charged particles of physics electrons, atomic nuclei, etc. have effective charges slightly less than their real charges, the ratio being about \(136/137\ .\) The effective charges are what one measures in all low energy experiments, and the experimentally determined value for \(e\) must be the effective charge on an electron, the real value being slightly bigger. In experiments involving energies of the order \(mc^2\) it would be the real charge, or some intermediate value of that charge which comes into play, since thepolarisationof the negative energy distribution will not have then to take in its full value. Thus one would expect some small alterations in the Rutherford scattering formula, the Klein Nishina formula, the Sommerfeld fine structure formula, etc. when energies of the order \(mc^2\) come into play. It should be possible to calculate these alterations approximately, since, although the ratio effective charge/real charge depends on the energy at which we assume the relativistic wave equation to break down, it does so only logarithmically, and varies by only about 12% when we double or halve this energy. If the experimenters could get sufficiently accurate data concerning these formulae, one would then have a means of verifying whether the theory of negative energy electrons is valid for energies of the order \(mc^2\ .\) I have not yet worked out the effect of magnetic fields on the negative energy distribution. They seem to be rather more troublesome than electric fields. With best wishes, and hoping to see you in September Yours sincerely, P. A. M. Dirac

In his letter Dirac had succinctly stated the physics entailed by vacuum polarization. The physical picture he sketched -- which is still accepted at present -- was elaborated by Uehling (1935) and by Weisskopf (1936).

In quantum electrodynamics the vacuum is not an empty medium but zero point fluctuations produce virtual electron-positron pairs. These are responsible for making the vacuum a dielectric medium that can be characterized by a dielectric constant, \(\epsilon\ .\) This dielectric constant is distance dependent: When a bare charge \(e(0)\) is introduced into the vacuum the charge that is observed at a distance \(R\) is given by \(e(0)/\epsilon(R)\ ,\) that is, the observed charge is reduced by an amount \(\epsilon(R)\ .\) Uehling determined that the larger \(R\) the more screening occurred: the effective charge at \(R\) increases as \(R\) decreases. [Coleman and Gross in 1973 proved that all quantum field theories -- with the sole exception of non-Abelian gauge theories -- have effective charges that decrease as the distance increases.]

The picture can readily be generalized to the case of time dependent electric fields, in which case the dielectric constant becomes frequency dependent. Dirac indicated how this is to be done. In the presence of a time dependent electric field, the procedure Dirac had outlined in his Solvay report is ambiguous, because one cannot in general separate the solutions of the Dirac equation in the presence of the time dependent field into positive and negative energy states in an unambiguous fashion. Nonetheless Dirac found a way to resolve these ambiguities by exhibiting the characteristic singularities of the density matrix on the light cone. Shortly after the Solvay Congress Dirac wrote Bohr

I have been working at the problem of the polarization of the distribution of negative energy electrons, from a relativistic point of view. If I have not made a mistake, then there is just one relativistically invariant, gauge invariant treatment, which gets over all the difficulties connected with the infinites, to the accuracy with which the Hartree Fock method applies... I have not yet seen whether this relativistic treatment leads to any kind of compensation of charge arising from the vacuum polarization ... I shall write to Pauli about this and hope it will satisfy his objections to the theory of holes.

In the paper in which he presented this new approach [Dirac 1934] the off diagonal density matrix \(R\) is defined by

\[ \Sigma_\mathrm{occupied} \Psi(x',t',k')\Psi^*(x'',t'',k'')= (x',t',k' |\mathbf{R}| x'',t'',k'') \]

where \((x',t',k'),(x'',t'',k'')\) are distinct sets of space-time spin variables, and the summation is over all occupied states. The spinors \(\Psi_n\) and its adjoint \(\Psi^*_{n}\) are one particle c-number Dirac wave functions in the presence of the external field, that are approximately determined by the Hartree Fock self consistent method. Dirac found that the characteristic singularities of \((k',x',t'*R*k'',x'',t'')\) depended only on \(x^2 = (x'-x'')^2\ ,\) and that these occurred when \(x^2 = 0\ .\) Dirac proposed that the infinities to which the singularities give rise be dropped, a process we now call charge renormalization.

Heisenberg in 1934 clarified and generalized Dirac's approach. Motivated by "the necessity to formulate the fundamental equations of the theory in a way that goes beyond the Hartree-Fock approximation method," Heisenberg introduced the operator

\[ R(xN,x'') = \Phi(xN)\Phi^{*}(x'') \]

where \(\Phi(x')\) and \(\Phi^{*}(x'')\) are quantized field operators satisfying the Dirac equation in the presence of the external field. The vacuum expectation value of \(R(x,x')\) reduces to the \((x't'*R*x''t'')\) considered by Dirac. Incidentally, Heisenberg was using a charge symmetric formalism that treated electrons and positrons on an equal footing, so the removal of all singularities was then straightforwardly effected by considering the limit \(x^2 = (x'-x'')^2 \to 0\) of the operator \(R(x',x'')\ .\) A *modern* version of Heisenberg's formulation can be stated as follows: Consider *reasonable* external fields for which the solutions of the Dirac equation for the field operator \(\Phi(x)\) are well behaved and free of divergences. The current operator can then be defined in terms of the operator \(R(x,x')\) as

\[ j_{\mu}(x) = - e \lim_{x6xN} Tr[(R(x,xN) - \text{subtraction terms})\gamma^0\gamma_{\mu}] \]

where the trace is over the Dirac spinor indices. The requirements that \(j_{\mu}(x)\) exist as a well defined operator on the state vectors of the system and that it be conserved, \(M^{\mu}j_{\mu}(x) = 0\ ,\) impose constraints on the subtraction terms. That these constraints could be satisfied is essentially what Heisenberg showed in his 1934 paper. In that same article Heisenberg also considered the extension of the formalism to the case that both \(\Phi(x)\) and \(A(x)\) are quantized, i.e., the full quantum electrodynamics. He noted that the subtraction terms for \(R(x,x')\) could be determined order by order in perturbation theory, but that, seemingly, the higher order terms contain divergences similar to that of the electron self energy divergences that the subtraction procedure could not remove. In particular he found that to order \(e^2\ ,\) the photon self energy diverged even before the limit \(x^2 \to 0\) was taken. Serber shortly thereafter noted that Heisenberg's result was a consequence of an incorrect canonical transformation: the gauge invariance of the theory guaranteed that the photon's mass was zero [Serber (1935)].

Heisenberg's last investigation of positron theory during the 1930s was a paper he wrote jointly with his student Euler. In 1933 Halpern had noted that hole theory implied that a photon could scatter off another photon: the scattering of light by light (or Halpern scattering as it was called during the 30s). Heisenberg set his students Euler and Kockel to calculate the cross section for Halpern scattering, and after a lengthy and difficult calculation they reported that to order \(\alpha^2\) the scattering matrix element was finite [Euler (1935); see also Achiezer (193~)]. Euler and Kockel had expressed their result in terms of an effective Lagrangian: the interaction between photons resulted from the presence in the Lagrangian of a gauge invariant term proportional to \(a (\mathbf{E}\cdot\mathbf{H})^2 - b(\mathbf{E}^2 - \mathbf{H}^2)^2\ ,\) \(a\) and \(b\) constants proportional to \(\alpha^2\) in addition to the free field Lagrangian \((\mathbf{E}^2 - \mathbf{H}^2)\ .\)

Heisenberg and Kockel's paper generalized Euler and Kockel's work. They found the effective Lagrangian to order \(\alpha^3\) induced by static, homogeneous, external fields when no real electron positron pairs could be produced. In their paper they also pointed out that the theory contained divergent vacuum self energy contributions (nowadays called *bubble* diagrams) that had to be subtracted. They also noted that in QED the fourth order contribution to Compton scattering diverged, as did the sixth order contribution to the scattering of light by light. The paper ended on a pessimistic note: hole theoretic QED must be considered provisional.

We have emphasized these works by Dirac and Heisenberg because they are the point of departure of Valatin's recasting of Heisenberg's subtraction prescriptions after World War II, the basis of the regularization procedures of Bogoliubov and Shirkov, and the origin of the operator product expansions. [Valatin, Bogoliubov]

Shortly thereafter Weisskopf simplified Heisenberg and Kockel's calculation and gave a thorough discussion of the physics involved in *charge renormalization* [Weisskopf 1936]. It is a particularly valuable aspect of Miller's source book that it makes available in English Weisskopf's important contribution to the understanding of the properties of the vacuum in QED (paper 8). This paper is undoubtedly one of the deepest and most insightful articles on QED written during the 1930s, a distillation of all the insights that had been accumulated up to that point. It included an unambiguous prescription on how to subtract the divergences encountered in hole theory when an external electromagnetic is present based on the assumption that "the following properties of the vacuum electrons are physically meaningless:

- the energy of the vacuum electrons in field-free space;
- the charge- and current-densities of the vacuum electrons in field-free space;
- a field-independent electric and magnetic polarizability of the vacuum, which is constant in both space and time." (p. 207)

The divergences are removed by a subtraction procedure that redefines the intensities of the external fields (what later became known as charge renormalization). Weisskopf's approach was a departure from straightforward perturbation theory^{15}. Drawing on the work of Euler and Kockel and that of Heisenberg and Euler, Weisskopf calculated the correction to the total energy density of the vacuum when a (constant) external electric and magnetic field is present ^{16}. In modern parlance this would be called calculating the effective Lagrangian.

## Infrared divergences

In contrast with the divergences stemming from the high energy quanta, divergences were encountered in calculating the cross-section for the emission of a photon in the scattering of a charged particle. Bloch and Nordsieck(1937) recognized that the difficulty was due to the approximate method of calculation. Any change of momentum by a charged particle will certainly result in the emission of low frequency photons, and the emission of such photons must be taken into account in the computation of cross-sections. As was evident from Fermi's formulation of the motion of free charged particles interacting with the electromagnetic field in his 1932 RPM article, a freely moving charged particle is accompanied by attached photons, corresponding to its Coulomb and Biot-Savart fields. Any deflection, i.e. any change in the direction of its motion will give rise to a different distribution of attached photons. The difference between these initial and final attached fields corresponds to the emitted photons during the scattering process. Bloch and Nordsieck (1937) and Pauli and Fierz (1938) indicated that if these features are taken into account and meaningful questions (that take into account the emission of low frequency photons in the scattering of charged particles) are asked -- finite answers are obtained.

Mention should be made of the one calculation that attempted to amalgamate all the previous insights in order to obtain a divergence free formulation of hole theory to order \(\alpha\ .\) At the suggestion of Bloch and Oppenheimer, Dancoff in 1939 investigated "to what extent the inclusion of relativistic effects modify the conclusions of Pauli and Fierz." The latter had treated the motion of the charged particles nonrelativistically. Dancoff calculated the radiative corrections to the elastic scattering of an electron in an external field the electron being described hole theoretically. Unfortunately Dancoff only included the effects of the transverse photons in calculating radiative corrections and omitted the contribution of the Coulomb interaction terms (repeating the oversight of Carlson and Furry in their 1934 computation of the self energy!). He thus obtained a divergent result and concluded that in hole theory a new type of divergence occurred in the radiative corrections to the elastic scattering of an electron by an external field. Dancoff was actually partially correct. He did in fact encounter a new type of divergence, divergences that would later be called vertex functions divergences. When combined with pieces of the self energy divergences and properly taking into account a wave function renormalization which Dancoff did these divergences cancel one another. Bloch and Oppenheimer had actually noted this cancellation for some of the diagrams and this was the point of departure of Dancoff's investigation.

The complete cancellation of divergences to order \(\alpha\) requires taking into account both the transverse photons and the Coulomb interaction terms. Lewis (1948) who redid Dancoff's calculation after the Shelter Island Conference in 1947, discovered Dancoff's mistake (see also Epstein 1948). Since no one before World War II checked Dancoff's calculation, his result only added to the "awareness of the basic difficulties" that weighed so heavily on the mind of all field theorists.

Why did no one redo Dancoff's calculation at the time? There were certainly many able physicists who could have done so, e.g., Nordsieck, Serber, Weisskopf. Why did the community not invest talent and energy to carry out this task? Had it done so the difficulties of QED might have been resolved much earlier. As noted earlier quantum electrodynamics played a crucial role in establishing the existence of the "mesotron" in 1935/6. This success helped shift the focus. Yukawa's 1935 paper, in which he had proposed, in analogy with QED, that nuclear forces were mediated by the exchange of massive bosonic quanta (rather than pairs of particles as had been suggested by Tamm (1934) and Iwanenko (1934) on the heels of Fermi's \(\beta\)-decay theory) had not been received with "immediate consent or sympathy" [Wentzel (1960)]. But after the "mesotron" was detected, it became immediately identified with Yukawa's nuclear force meson and Yukawa's paper became "the focus of universal attention" since "Yukawa had predicted its existence" [Wentzel (1960)]. Nuclear forces and meson theory was pushed to the center of theoretical interest; and the explanation of the electromagnetic properties of the nucleons and those of the deuteron, such as their magnetic dipole moments and electric quadrupole moment, became the challenging problems [Heitler (1939); Møller Rosenfeld (1940); Schwinger (1942)]. Similarly, the determination of the characteristics that mesotrons of spin \(0, 1/2, 1, \cdots\) would manifest in cloud chambers, was actively investigated. And a wealth of new experimental data was being produced.

That is not to say that there were no intriguing experimental results available to the theoretician interested in testing QED. By 1938, the experimental data on the spectrum of hydrogen suggested that the \(2S_{1/2}\) and \(2P_{1/2}\) levels were not degenerate, contrary to the predictions of the Dirac theory. Both Houston (1937) and Williams (1938) had found these departures from the Dirac theory. Pasternack (1938), in fact, indicated that these deviations could be interpreted as an upward shift of the \(2S_{1/2}\) level by 0.03 cm approximately 1000 megacycles, relative to the \(2P_{1/2}\) level. Stimulated by these findings, Fröhlich, Heitler and Kahn (1939,40) calculated (incorrectly, as it turned out) the form factor of the proton according to a version of meson theory and obtained an upward shift of about \(0.03 cm^{-1}\) in agreement with Pasternack's requirements. It was incidentally Willis Lamb (1939,40) who pointed out the mistake in their calculations! But the ambiguous nature of the experimental data should not be underestimated. The data were at the limit of the resolution of the optical apparatus then available and difficult questions concerning the assumptions on which it was based could not easily be resolved (e.g., was the H gas in the discharge tube in thermal equilibrium as was assumed in imposing the usual Doppler broadened shape to the observed lines). Thus it is not surprising that these deviations from the predictions of the Dirac equation should be taken it *cum grano salis* by the theoretical community. Any one faced with the staggering complexity a hole theoretic computation entailed would have wanted a firmer experimental inducement to undertake such calculations.

But probably the most important factor in delaying the completion of the subtraction program and its application to render QED finite were the events surrounding the outbreak of World War II. With the ominous developments in Nazi Germany, after the discovery of the fission of uranium the leading theoretical physicists in England and the United States turned their attention to the problems implied by the possibility of a chain reaction. In particular, this was the case for Weisskopf, one of the most likely person to have made a relativistic quantum electrodynamical calculation of the spectrum of hydrogen before World War II. With the fall of the Low Countries and of France in the spring of 1940 scientific contact with European colleagues on the Continent except for those in Switzerland essentially ceased. Thus Kramers' insights had to await the 1947 Shelter Island Conference to be appreciated by the theoretical community in the United States. That there is truth in this explanation is borne out by the following: In the fall of 1946, Weisskopf, who had just come to MIT, gave Bruce French, then a graduate student, the problem of calculating the \(2S_{1/2} - 2P_{1/2}\) level splitting in hydrogen. This was well before the experiment of Lamb had become known! Oppenheimer's original suggestion that differences in energy levels might prove finite was to be the basis of the calculation. Weisskopf's 1939 analysis of the self energy of the electron in hole theory had made the chances for the success of such a calculation much greater.

If the boundaries of knowledge lie between the possible and the unthinkable, between sense and nonsense, part of the reason that the workers of the thirties did not overcome the divergence problems was that they were too ready to focus on the unthinkable i.e., on radical and revolutionary departures. Progress, it would turn out, could be achieved by a conservative stand: by taking seriously the successes achieved by charge renormalization procedure in the vacuum polarization problem and using the insights gleaned from the work of Kramers and Pauli and Fierz regarding mass renormalization to circumvent the self energy divergences.

## Mesotrons and Meson theories

In 1935 Hideki Yukawa published a paper in which he proposed a field theoretical model to account for the nuclear forces. In Yukawa's theory the neutron-proton force was mediated by the exchange of a hitherto unknown scalar particle between the neutron and proton, with the mass of the scalar particle so adjusted as to yield a reasonable range for the nuclear forces. Shortly after the Cal Tech cosmic ray physicists Carl Anderson and Seth Neddermeyer had given evidence for the existence of a new type of particle in the penetrating component of cosmic rays, Oppenheimer and Robert Serber in 1937 published a short note in the Physical Review pointing out that the mass of the newly discovered particle, specified a length which they connected with the range of the nuclear forces as had been suggested by Yukawa. Oppenheimer and Serber's note drew the attention of American physicists to the *meson theories* of nuclear forces that Yukawa, Ernest Stückelberg and Gregor Wentzel had advanced [See Kemmer 1985]. The existence of this *heavy electron* -- which existed in both a positive and a negative variety -- was authenticated by its direct observation in a cloud chamber by Curry Street and Edward C. Stevenson, who also determined its mass (-150-220 electron masses) from measurements of the ionization it produced, and from the curvature of its track in a magnetic field. Its lifetime was estimated to be about \(10^{-6}\) sec.
By 1939 Bethe could assert that "it was natural to identify these cosmic ray particles with the particles in Yukawa's theory of nuclear forces."

The importance of Yukawa's work is that he was the first to extend Fermi's field theoretical approach, and to cement the notion that the forces between observable particles are generated by the exchange of quanta. Furthermore note the comment made by Fröhlich in 1985:

In 1937, Heitler and I were interested in understanding the deviation of the magnetic moment of the proton and the neutron from the Dirac moment in terms of virtual absorption and emission of aheavyparticle satisfying Bose statistics. One of the main problems arising here concerned the symmetry of the particle. Only later did we realize that Yukawa's general idea would involve this possibility. (Frohlich 1985, p.5)

Fröhlich's remark indicates that by 1937 part of the metaphysics of *elementary particle* physicists was to postulate the existence of hitherto unobserved fields and their associated quanta to account for empirical phenomena. This is in sharp contrast with the reticence on the parts of theorists to introduce new particles in the early 30s. Recall Pauli's handling of his suggestion that in \(\beta\)-decay a neutrino is emitted, and Dirac's initial hopes that *holes* corresponded to protons.

In September 1940 the University of Pennsylvania celebrated its bicentennial anniversary with a week long series of symposia. The theme of the physics symposium was "Nuclear Physics"^{17} and the speakers included Enrico Fermi, Gregory Breit, Isadore Rabi, Eugene Wigner, Robert Oppenheimer and John Van Vleck. The talks surveyed the current state of knowledge in the field and highlighted the problems to be solved. Coming a year after the outbreak of World War II and a few months after the fall of France, it was the last time these physicists would address problems in "pure" physics. All of them, in fact, were already deeply immersed in work that would culminate in the assembly of a uranium and a plutonium atomic bomb.

Fermi spoke about the experiments he was performing at Columbia on the reactions produced by the neutron bombardment of heavy elements and in particular, on the analysis of neutron induced fission in uranium. In his talk Breit reviewed what had been learned about the two-body nuclear forces from neutron-proton and proton-proton scattering. For his contribution to the symposium Wigner updated the paper he had published in 1937 on nuclear binding energies in the light "the new experimental data that had greatly enriched the field over the past four years." Rabi in his presentation described the experiments that were being carried out at Columbia using molecular beams to measure the spin and magnetic moment of the lighter nuclei and reported on the most recent data for the quadrupole moment of the deuteron. (\(2.73 x 10^{-27} cm^2\)). He also reviewed the implication that the existence of the deuteron quadrupole moment had for the neutron-proton force and noted that the 1938 paper of Fröhlich, Heitler and Kemmer, and a more recent one by Bethe (1940), had indicated that spin one meson theories could explain not only the deuteron's binding energy and its level structure, but also its quadrupole moment. Oppenheimer talked about "The Mesotron and the Quantum Theory of Fields" and undertook to assess the current state of knowledge and use of meson theory. It was a natural task for him to undertake, for at the time Oppenheimer was at the center of the researches carried out in meson theory in the United States.^{18}

From 1937 until Pearl Harbor, Oppenheimer and his students were actively engaged in investigating possible descriptions of the newly discovered cosmic ray meson and its connection to nuclear forces. Two of Oppenheimer's students -- Lamb and Schiff (1938) -- had found that Yukawa's theory in its original form (charged mesons of zero spin) gave the wrong sign for the interaction potential in the deuteron, i.e. a repulsion instead of an attraction in the \(^3S\) state. This could be remedied if it was assumed that the meson had spin one -- and Yukawa, Sakata, Taketani, and Kobayasi in Japan, and Frohlich, Heitler and Kemmer, and H. Babbha in Great Britain, developed this approach. A spin one vector meson theory gives rise to a spin dependent tensor force. However, any theory involving only charged mesons will give no force between like nuclear particles in lowest approximation (that is with the emission and absorption of a single meson.) To second order, Frohlich, Heitler and Kemmer found a strong repulsive force at small distances. This contradicted the experiments of Tuve, Heydenburg and Hafstad at the Carnegie Institution in Washington which indicated that the force between two protons is strongly attractive and very nearly equal to that between neutron and proton in the singlet state. A force between two like particles can only be obtained with neutral mesons. Kemmer developed a theory of nuclear forces in which neutral and charged mesons occur in a symmetrical way and thus explained in a natural way the equality of the forces between like and unlike nuclear particles. A characteristic of the symmetrical theory is the specific exchange character of the nuclear forces which made it possible to explain saturation of the nuclear forces simply in a manner analogous to the homopolar chemical bond.

An alternative way of explaining charge independent forces is to assume interactions with neutral particles only. Then the charge of the nuclear particle (i.e. whether it is a neutron or a proton) becomes irrelevant and the equality of the forces follows immediately. This alternative was explored by Bethe and found to be successful when applied to the two-body problem -- but suffered from the disadvantage that must appear in any purely neutral meson theory, namely that

a) there was no connection with the cosmic ray mesons that had been observed and

b) no explanation of \(\beta\)-decay nor of the anomalous magnetic moments of neutron and proton.

Towards the end of the thirties most of the research activities in Oppenheimer's group centered on meson theory. All aspects of meson theories -- and in particular, their consequences for cosmic rays physics, nuclear forces and \(\beta\)-decay -- were thoroughly investigated by Oppenheimer and his students. Incidentally, two such investigations -- one by Willis Lamb, and the other by Julian Schwinger -- were to prove of great importance in the post World War II developments in quantum electrodynamics.

Willis Lamb had enrolled as a graduate student in physics at Berkeley in the fall of 1934. During his second year at Berkeley, Lamb spent a great deal of time calculating the nuclear forces predicted by various field theories. His first investigation consisted in exploring the nuclear forces generated by the Fermi interaction, that is, those produced by the exchange of pairs of electrons and neutrinos between nucleons. The resulting nucleon nucleon potentials were highly singular "and then only if infinite integrals were interpreted with the help of convergence procedures" [Lamb in L. Brown et al. (1983), p. 316]. After the appearance of Yukawa's papers, Lamb worked on the nuclear forces generated by Yukawa's meson, and in particular on the deviations from Coulomb's law due to mesonic effects. The question of the validity of Coulomb's law became a recurrent theme in Lamb's researches [Lamb (1960)]. In 1939, Lamb became involved in a controversy with Fröhlich, Heitler, and Kahn (1939a,b) about their meson field theoretic calculation that indicated that the electric field around a proton should deviate from a pure \(1/r^2\) Coulomb field. These theorists were actually trying to account for the deviations from the predictions of the Dirac equation for the spectrum of hydrogen that Houston (1937) and R.C. Williams (1938) had observed. They thought that in addition to the attractive Coulomb interaction between an electron and a proton they could get a short range repulsion by the mesotron charge distribution around the proton. Lamb (1939,1940) argued that their result was possible only because they were working with a perturbative solution of a theory plagued with divergences. Although the differences were never resolved, the discussion was useful because it kept Lamb "thinking about the hydrogen fine structure." It also made him aware of "the suggestion of Simon Pasternack (1938) that Houston's spectroscopic data could be interpreted in terms of a short range repulsion between electron and proton" (Lamb 1983). Later at Columbia, Lamb was to carry out the definitive experiment on the fine structure of hydrogen.

The other investigation carried out at Berkeley before the war that had a crucial bearing on the post war QED work was that of Schwinger. Schwinger had gone to Berkeley in the fall of 1939 as an NRC fellow, and stayed there for two enormously productive years. While there he worked on a wide range of subjects, all of which had a common goal, namely to understand the nuclear forces better. The investigations ranged from phenomenological analyses of the empirical data on the deuteron and light nuclei to extensive field theoretic calculations using differing spin and charge assignments for the *mesotrons* involved. While still at Columbia, Kemmer's 1938 paper on a *charge-independent* meson field theoretical model of the nuclear forces had made Schwinger aware of the possibility of the presence of tensor forces in the neutron proton interaction. An analysis of the electromagnetic properties of the deuteron when tensor forces are present led him to predict the existence of the deuteron's quadrupole moment before it had been measured by Kellogg, Ramsey, Rabi and Zacharias [Kellogg 1939,1940]. Tensor forces, and the kind of *mesotrons* and meson nuclear couplings that could give rise to them, became a central focus of the Schwinger's investigations at Berkeley. The piece of research that was of great importance after the war when Schwinger worked on QED was his Berkeley work on strong coupling mesotron theory [Oppenheimer (1941), Schwinger (1970)], through which he gained experience in using canonical transformations to extract the physical consequences of a field theory.

The *mesotron* that had been identified by Neddermayer and Anderson, and by Street and Stevenson exhibited weak nuclear interactions in its passage through the atmosphere. Field theorists were thus trying to find a meson theory that yielded strong nucleon nucleon forces yet a small meson nucleon cross section. Heitler (1940) and Bhabha (1941) independently indicated how this could come about. They observed that the cross section for the scattering of a neutral scalar meson on a nucleon would be quite small, because the direct and crossed Born terms very nearly cancel. For a charged meson the cross section is large, because only one of the Born terms exists. They pointed out that, if in addition to the nucleons there existed low lying excited states of the nucleon "isobars" with charge \(+2e\) and \(-1e\ ,\) then there would be two Born terms which could nearly cancel, and thus produce a small cross section as in the neutral case. Somewhat earlier, Heisenberg (1939) had noted a different mechanism. He had calculated the scattering of a neutral vector meson coupled to the spin of a nucleon and had found that the large Born approximation result was greatly suppressed by "reaction effects." He observed that the self field of the nucleon, that is that part of the meson field which is attached to the nucleon and carried with it, was responsible for an increased inertia of the spin motion, and resulted in a cross section of the order of \(a^2\) where \(a\) is the assumed radius of the nucleon.

The next important step was taken by Wentzel, who in 1940 showed that the simplest nontrivial static model, that of a charged scalar meson field coupled to a fixed scatterer, could be solved quantum mechanically in the limit of large coupling constant. He found that the solution exhibited isobars; however, these isobars did not result in a particularly small cross section because their excitation energy was not small enough. Oppenheimer and Schwinger (1941) extended Wentzel's strong coupling calculations to the case of neutral pseudoscalar mesons. They found that Heisenberg's (1939) classical result for the scattering was reproduced, and noted that the solution gave rise to isobars that had been overlooked by Heisenberg and reduced the cross section for meson nucleon scattering by the Heitler Bhabha mechanism. Stimulated by Wentzel's pioneering work on strong coupling theory, Schwinger gave a more extensive treatment of the strong coupling limit of the charged scalar mesotron field. He observed that the large coupling should serve to bind mesotrons in stationary states around the nucleon and give rise to isobaric states, which he exhibited by using a series of canonical transformations [Schwinger (1970)].

It is with this background in mind that I return to Oppenheimer's paper at the Pennsylvania bicentennial celebration. Oppenheimer judged his report as "negative" for his purpose was to explain why

the theory of the mesotron, and more generally, the quantum theory of fields, has failed so completely to deepen our understanding of nuclear forces and processes, has left almost totally unanswered the many questions earlier speakers have put to it; and to try to explain, too, why in spite of this the quantum theory of fields still seems to me a subject worth reporting on at all.

He noted that the discovery of the mesotron, although it sharpened all the difficulties of field theory, has also

given us some confidence that the fundamental ideas of field theory are right, for the mesotron was a prediction, very general and qualitative it is true, of this theory.

Oppenheimer reviewed the foundations of the quantum theory of fields and stressed the importance of Bohr and Rosenfeld's analysis of the measurability of the field operators describing the electromagnetic field^{19}. They had shown that the uncertainty relations required by QED for the values of the various components of the field strengths, averaged over finite space time regions, are verifiable in principle. and that the quantum mechanical description of electromagnetic fields produced by classically describable charge and current distributions "was a consistent theory free of all contradiction." Oppenheimer believed that it was extremely important to extend these considerations to the case when the atomic constitution of the test bodies is made an essential feature of the argument, i.e. when the matter fields are quantized.

Oppenheimer went on to describe one of QFT's greatest successes namely, the spin-statistics theorem due to Pauli -- that is, the proof that particles of half integral spin must obey the exclusion principle; particles of integral, or zero, spin, must obey Bose statistics -- and indicated that the discovery of the mesotron has led to investigations inquiring whether there were physical grounds that could be advanced to limit further the possible types of particles that can be found in nature. Several of his students and postdoctoral fellows -- Snyder and Schiff and Rarita and Schwinger -- had, in fact, proposed to exclude those particles for which the electromagnetic self-energy was not only infinite but "too infinite", or to reject those for which the field equation describing them cannot be solved in the presence of certain simple external electromagnetic fields -- such as a mesotron of spin one in a Coulomb field, or any charged particles of spin greater than one in any electromagnetic field whatsoever. Oppenheimer however warned that these arguments might be "illusory" and only signified that "the abstraction of considering the external field as given [might] often be an impermissible over-idealization."

Oppenheimer then turned to a review of Yukawa's theory and its extension to the case where the mesotron has spin one and is described by the Proca equation. He noted that this theory "rather arbitrarily gives the mesotrons a unit magnetic moment,... does not describe nuclear forces, but it is fashionable at the moment to believe it." Some support for the theory came from the work of his students, which indicated that a charged mesotron of spin one and unit magnetic moment would produce about the right number and energy distribution of very high energy electron secondaries to account for the observed cosmic rays bursts. But Oppenheimer cautioned that the experimental values are neither complete nor reliable, and that "the ambiguities of theory bedevil these comparisons." Thus a particle of spin one and magnetic moment zero would be in better agreement, and a particle of spin 1\2 moment one almost as good.

But Oppenheimer's most incisive comments were devoted to the implications of the fluctuation phenomena that Bohr and Rosenfeld had elucidated in their analysis of the measurability of the quantized electromagnetic field operators. He pointed out that the characteristic values of an interaction energy which involves three fields at the same point averaged over a little region of space of dimension \(d\) about that point is proportional to an inverse power of \(d\ ,\) and therefore becomes infinite as one lets the dimension of the region go to zero. He thus concluded

that the expression usually written down for the interaction energy between fields are meaningless, in that the problem defined by them has no solutions. But it is just these interaction terms... used in a special restricted way, that have given us most of the successful predictions of [QED]. How is one to understand this?

One possible way was

to say that the expressions we use for the coupling are not quite right and that when just the right ones are found, and the correct values for charges and masses inserted, and moments and spin, then the present formalism will give a total energy that is finite , and all the troubles will disappear because the infinite terms will cancel.

But Oppenheimer did not know how to do this. A second possibility was to give up the notion of point coupling. This however raised the question --"urgent for the theory of nuclear forces, of what is the nature of this extended source distribution." The third possibility -- in the nuclear force problem-- was to treat the trajectories of the neutron and protons classically, but accept the largeness of the coupling between nucleons and mesons -- " but " Oppenheimer noted, "we are a long way from knowing how to handle [large couplings.]"

He concluded his presentation with the following assessment:

The largeness of the couplings, and Yukawa's relation between nuclear range and mesotron mass, seem to us about the only fairly sure conclusions that we have.

The war interrupted all pure physics activities in the United States. Except for Pauli, who spent the war years at the Institute for Advanced Study in Princeton, and some of his associates who were foreigners, essentially no meson theoretic work was carried out in the United States during the war^{20}.

The state of affairs in "elementary particle" physics just before the entry of the US in World War II was summarized by John Archibald Wheeler [1911- ] in an important address delivered at a joint meeting of the National Academy of Sciences and the American Philosophical Society in the fall of 1945. Wheeler observed that the experimental and theoretical researches of the 1930s had made it possible to identify four fundamental interactions: a) gravitation, b) electromagnetism, c) nuclear (strong) forces, d) weak decay interactions. But even though one could distinguish between four kinds of interactions each having distinctive coupling constants-- it was clear by the end of the decade that to explain the electromagnetic properties of neutrons and protons, meson theoretic considerations had to be invoked. This interlinkage and interconnectedness of the weak, electromagnetic and strong nuclear interactions became a permanent feature of quantum field theoretical explanations.

## Epilogue

The salient features that emerge from the above cursory history are the following :

1) There is a clear-cut periodization:

1927-1933 | Formulation and elaboration of QED |

1933-1947 | Formulation and elaboration of QFT |

That is of course not to say that there are no further elaboration of QED after 1933. There of course are (see the section labelled Periodization). Rather we want to emphasize the radically new element introduced by Fermi's \(\beta\)-decay paper in 1933. The common features of (relativistic) QED and QFT were that all calculations higher than second order contained divergences.

2) The gradual adoption for QFT of the philosophical standpoint that the relevant features to be accounted for are experimentally established quantities (such as cross-sections, energy levels ...) Hence the gradual abandonment of the metaphysics of the Lorentz model of extended charge particles and the move towards Kramers' viewpoint regarding descriptions in terms of the experimentally determined parameters instead of the parameters appearing in the Lagrangian. This culminated with Heisenberg's S-matrix program, in which he formulated a relativistic description of interactions that was not based on a state vector whose evolution is determined by a Hamiltonian with interaction terms, but one which limited itself to giving the results of scattering experiments.

3) the adoption of a viewpoint that postulated new fields to

a) either account for observed properties (nuclear forces, magnetic moments of nucleons, quadrupole moment of the deuteron....)

b) soften the \(r\) dependence of the nuclear forces for \(r \to 0\) (Møller-Rosenfeld)

c) compensate electromagnetic forces by forces due to the new fields, and thus attempt to alleviate the self-energy divergences (at least in second order) (Stuckelberg 1939, 1941), Bopp (1940).

4) The role of symmetries becomes more prominent throughout the decade: SU(2) was introduced by Heisenberg in his description of neutrons and protons, and by Kemmer to formulate his charge-symmetric theory to account for the equality of nn, pp, and np forces. Similarly, the importance of maintaining gauge invariance in QED calculations becomes apparent. (Serber 1936, Weisskopf 1936).

Regarding the divergence difficulties, and the various subtraction schemes that were formulated during the 1930s (see Pais ) the recent comments by Dyson are applicable to that period too:

All through its history quantum field theory has had two faces, one looking outward and one looking inward. The outward face looks at nature and gives us numbers that we can calculate and compare with experiments. The inward face looks at mathematical concepts and searches for a consistent foundation on which to build the theory. The outward face shows a brilliantly successful theory, bringing order out of the chaos of particle interactions, predicting experimental results with astonishing precision. The inward face shows us a deep mystery. After seventy years of searching we have found no consistent mathematical basis for the theory. When we try to impose the rigorous standards of pure mathematics , the theory becomes undefined or inconsistent. From the point of view of the mathematician the theory does not exist. This is the great unresolved paradox of QFT. (Dyson in Mitra 2000, p. viii-ix)

This great paradox saw the light of day during the 1930s.

## Periodization

A periodization of QFT during its first decade can readily be given:

1927-1933 | Formulation and elaboration of QED |
---|---|

1927 | Dirac's QED ( Pro. Royal Soc. Articles) |

1928 | Jordan-Klein, Jordan-Wigner |

1929 | Heisenberg and Pauli - Quantum Theory of wave fields |

1930 | Fermi's QED ; lectures at Michigan (RMP article) |

1931 | Dirac's hole theory |

1932 | Dirac, Fock, Podolsky: multiple time formulation of QED |

1932 | Bethe and Fermi -- interaction generated by exchange of quanta |

1933/4 | Dirac: Vacuum polarization |

1933 | Hole theoretic calculations of pair production, Bremsstrahlung, ... |

1934 | Weisskopf/Furry-Carlson : logarithmic divergence of electron's self-energy in hole theory |

1934 | Halpern, Euler, Kockel, Akhezier, Delbruck, Heisenberg, ... :scattering of light by light |

1936 | Serber: emphasis on gauge invariance to alleviate divergence difficulties in QED |

1936 | Weisskopf on vacuum polarization |

1936 | Accomplishments and difficulties : Heitler's Quantum Theory of Radiation |

1937 | Bloch and Norsieck: Infrared divergences |

1938 | Kramers' formulation of the interaction of charged particles with the radiation field |

1933-1947 | Formulation and elaboration of QFT |

1933 | Fermi's theory of \(\beta\)-decay |

1934 | Pauli-Weisskopf: quantization of the charged scalar field |

1935 | Yukawa's theory of the mesotron responsible for nuclear forces |

1936 | Dirac higher spin equations |

1936 | Pauli Connection between spin and statistics |

1938-41 | Study of the nuclear forces stemming from various meson theory |

1938-9 | Møller-Rosenfeld |

1940 | Bopp, Stuckelberg: cancellation of divergences by introducing new fields |

194X | Heisenberg: S-matrix theory |

1943 | Tomonaga : QFT on space-like surfaces |

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## Endnotes

^{1} This possibility is first mentioned in his 1930 paper dealing with the self energy of the electron.

^{2} An external electromagnetic field will produce (virtual) pairs of electrons and positrons, which polarize the vacuum. A charged particle will surround itself with the charged particles of opposite charge created by its field. The theory predicts an infinite polarizabilty for these effects.

^{3}Thus the opening sentences of Brown and Rechenberg=s study of the origin of the concept of nuclear forces are the following:

This book is a study of the history of fundamental theories of nuclear forces, from 1932 to 1950 ..., beginning with the discovery of the neutron and ending with the discovery of the neutral pion. By "fundamental theory of force" we mean ( in a microscopic sense) that the force arises from the exchange of quanta. (L.M. Brown and Rechenberg 1996, p.ix)

^{4} Heisenberg actually estimated both the strength and the spatial dependence of the resulting charge exchange potential \(J(r)\) using the Fermi constant obtained from \(\beta\)-decay data:

\[ J(r).mc^2 (10^{-14}/r)^5 \]

For \(r\) of the order of the range of nuclear forces, \(10^{-13} cm\ ,\) \(J(r)\) is too small by 6 orders of magnitude.

^{5} In the Introduction Miller points to two alternative "scenarios" for progress in QED during the 1930s:"one emphasizing rationalism coupled with conceptual analysis; the other emphasizing empiricism. The first -emerges from the primary papers and their historical analysis in this volume." (p. xv)

^{6} These calculations are presented in Heitler's Quantum Theory of Radiation. (Heitler 1936)

^{7} Quantizing meant promoting the classical variables into operators satisfying the commutation rules \([q,p]= \le i\ .\)

^{8} See also Talbot (1978) and in particular Volume 7 of Niels Bohr, Collected Works, which contains Bohr=s correspondence in connection with the Bohr-Rosenfeld paper.

^{9} e.g Heisenberg 1934, Morrison 1939.

^{10} See in particular paper 8 in MIller=s source book, Weisskopf's "The electrodynamics of the vacuum based on the quantum theory of the electron."

^{11} See also Milonni (1984).

^{12} There were many attempts in this direction. For example, in his paper on fundamental length Heisenberg points to the work of March (1937).

^{13} For example, one can assert that the charge renormalization brought about by vacuum polarization effects in the presence of a classically described external electromagnetic field was understood during the thirties.

^{14} The self-energy calculations by Heisenberg and Fermi were carried out before Dirac had formulated his hole theory wherein all the negative energy states of the Dirac equations are filled, and holes in that distribution would correspond to positrons. The calculations by Oppenheimer and by Waller had indicated that to lowest order of perturbation theory the self energy of an electron diverged linearly in a pre-hole theory.

^{15} Because of the smallness of the fine structure constant, \(e^2/hc\ ,\) in the usual analysis of the structure of the vacuum in QED one would start with an unperturbed vacuum state consisting of a "free" (noninteracting) electromagnetic field and of noninteracting electrons filling the negative energy levels of the Dirac sea.

^{16} Such an electric and magnetic field affects the energy levels the electrons in the Dirac sea.

^{17} Nuclear Physics. Philadelphia: University of Pennsylvania Press 1941.

^{18} Oppenheimer's researches during the 1930s had dealt with subjects at the frontiers of physics: cosmic rays, quantum electrodynamics, nuclear physics, astrophysics and cosmology. All his investigations and that of his students no matter how esoteric or formal always had the experimental situation in mind. Often they were motivated by experimental findings at Cal Tech and at Berkeley; e.g. his theory of pair production; his explanation of cosmic ray showers as being initiated by electrons emitting x rays, which x rays in turn produce electron positron pairs in the electric field of the atomic nuclei of the atmosphere; and the attempt by his student, Edwin Uehling, to explain the deviations in the spectrum of hydrogen from the predictions of the Dirac equation as a result of vacuum polarization. Oppenheimer's letters to Bethe, Pauli, Uhlenbeck and Wigner attest to his mastery of all aspects of nuclear and "high energy" physics, ranging from detailed knowledge of nuclear energy levels, nuclear reaction cross sections, beta ray spectra, to the most recondite calculations in meson theory and quantum electrodynamics.

^{19} In the spring of 1933, Bohr had visited Berkeley and lectured on his work with Rosenfeld on the problems of the measurability of the electromagnetic field. Furry recalled that these issues "were taken very seriously".

^{20} See Pauli's "Meson Theory" the set of lectures he gave at the MIT
Radiation Laboratory toward the end of the war, and his RMP article in 1946 for a summary of these activities.

## Further reading

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## See also

gauge invariance, gauge theories, quantum electrodynamics, renormalization, second quantization