Alexander S. Blum and Dean Rickles

Quantum Gravity in the First Half of the Twentieth Century

Series Editors
Ian T. Baldwin, Gerd Graßhoff, Jürgen Renn, Dagmar Schäfer, Robert Schlögl, Bernard F. Schutz

Edition Open Access Development Team
Lindy Divarci, Samuel Gfrörer, Klaus Thoden, Malte Vogl, Dirk Wintergrün

Editor-in-chief
Matteo Valleriani
Max Planck Institute for History of Science, Berlin

Other Editors
Stephen P. Weldon
Department of History of Science, University of Oklahoma
Esther Chen
Library of the Max Planck Institute for the History of Science, Berlin
Kerry V. Magruder
History of Science Collections, University of Oklahoma Libraries
Anne-Laurence Caudano
History Faculty, The University of Winnipeg
Massimiliano Badino
Facultat de Ciéncies, Universitat Autònoma de Barcelona
Robert G. Morrison
Department of Religion, Bowdoin College

First published 2018
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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.

Max Planck Research Library for the History and Development of Knowledge
Sources 10

Preface

Bryce DeWitt recalls having a conversation with Pauli in 1949, at the Institute for Advanced Study, in which Pauli asked him what he was working on,[1] to which DeWitt responded: “trying to quantize the gravitational field.” Pauli was something of a veteran of quantum gravity by this time. His response, after shaking and nodding his head a few times (“die Paulibewegung”), was: “That is a very important problem—but it will take someone really smart!”.[2] More than half a century since this conversation, and with the work of many genuinely “really smart” people (including several geniuses), the problem has, of course, not yet been resolved. Yet still the field has accumulated a rich and interesting past that has yet to be properly studied. It has become customary to mark a certain stage of maturity of a theory by producing a volume of sources of early papers from which that maturity emerged.

Though quantum gravity has not yet achieved full scientific maturity, it has at least achieved chronological maturity, with almost a century of struggle behind it. Therefore, we feel it is entirely appropriate to treat this old timer with some respect, of which it has not received all that much from the history of physics. A volume providing a historical overview, after so long without one, can be beneficial to the current and future generation of physicists working on the problem, in order to see how far research on the problem (and the way the very problem itself is conceptualized) has come—this further provides a fresh perspective on what still remains to be done. It might point to further refinements of how we understand the problem so that it can finally be resolved.

As Julian Schwinger pointed out in the preface to his own collection of papers from the history of quantum electrodynamics,[3] any such selection of sources is bound to reflect the particular viewpoint of the editor(s). Following Schwinger, we briefly describe our “selection process.” Despite the fact that the period we cover spans only 35 years, it was necessary to be fairly brutal in rejecting papers for which a case for inclusion could easily be made. Likewise, it is likely that cases could be made for excluding many of the papers we decided to include. The point is, the sources chosen are an imperfect reflection of the development of a field, and one important reason for this is that “the field” in question has always been somewhat slippery and hard to define, but especially so in its earliest phases of development. One can’t, for example, point to particular phenomena that the theory will describe since any such phenomena would be experimentally and observationally very remote. Moreover, in the earliest phases of research, the ingredient theories (general relativity and quantum mechanics) were themselves still being worked through and, in the case of the latter, were not properly formulated for some time (as Part I indicates). Thus one finds the definition of the problem of quantum gravity is non-stationary on account of being largely at the mercy of wider developments in quantum mechanics and general relativity in our chosen time period—one sees this especially clearly in Part II, but it is really a general feature.

Firstly, let us explain the period we have restricted our sources to, namely <1950. This is a relatively short snapshot of history, but it has the advantage of revealing the steps taken before the rather dramatic explosion of work in the 1950s—this explosion was due to a variety of factors beyond internal advances in physics, including the emergence of new schools of research in general relativity (especially Peter Bergmann’s, Hermann Bondi’s, Alfred Schild’s, Leopold Infeld’s, and John Wheeler’s) and new sources of funding (especially the NSF, the ONR, and other military and philanthropic programmes).[4] We also find that the pre-1950 research already includes many of the main lines of attack and the main general arguments for (and against) quantization. Focusing in on the very earliest period sharpens the physical intuitions behind the various choices (of formalism, terminology, and more) that have since been assimilated or forgotten.

We will now go on to explain why there are papers in here (many, in fact) that are not strictly “quantum gravitational”. One can quite usefully think of the development of research programmes (and questions) in a field in terms of “evolutionary trees.” Pursuing this in the case of organisms eventually leads one outside of the species of interest. Or one might find branch points, in which now divergent organisms converge onto a common ancestor. Likewise with the evolution of a field of inquiry. Since this is a “sourcebook”, rather than a straight history, we are guided by our present day theories and approaches, and so are more concerned with tracing back various ancestors. Some of these ancestors look like the present day approaches, and others don’t. But regardless of which is the case, they have nonetheless been involved in the development of the present approaches. Hence, we have often erred on the side of being too liberal where ideas that originated in a slightly different context were nonetheless incorporated into quantum gravity research at some later date.

Quantum gravity is, of course, yet to be articulated in any final, agreed upon formulation. As alluded to above, with new developments in physics, the quantum gravity project would attempt to avail itself of some potentially relevant feature—it is, thus, a “parasitic” enterprise for much of its early history: wave mechanics, spin and the Dirac equation, neutrinos, the discovery of new forces, mesons and cosmic rays, ... . All of these and more were immediately taken up as of potential relevance in quantum gravity’s definition and domain. Parts I and II study the ways in which quantum gravity was studied in its embryonic and infancy stages. In cases where it is not parasitic, it is viewed not so much as a problem in its own right, but as an interesting case study, or else a resource to cure problems in field theory more generally (the “more serious” business of physics). For example, in 1938 one can find Born writing that there “seems to be a general conviction that the difficulties of our present theory of ultimate particles and nuclear phenomena (the infinite values of the self energy, the zero energy and other quantities) are connected with the problem of merging quantum theory and [general] relativity into a consistent unit”.[5] Parts III and IV cover such aspects. Part III also deals more generally with the direct quantization of general relativity using techniques that had been applied to the electromagnetic field.

What is interesting about these early papers, in terms of the “shut up and calculate” narrative that is often told about physics after the war, is that the papers very rarely step into conceptual waters. One can find none of the preoccupation with the status of observables, the existence of space and time, the meaning of diffeomorphism symmetry, and other such foundational problems that spring up in the 1950s—almost immediately where the papers in this volume stop, in fact. To a certain extent this later development had to do with physicists stepping back and considering the classical theory of general relativity more carefully from a physical point of view than had been done previously. The reasons for this are clear: the standard techniques faced technical problems of their own. There was a dawning recognition, towards the end of the first half of the twentieth century, that gravity was simply not like other forces.[6] This recognition brought with it the idea that the problem of quantum gravity will most likely not be resolved through a purely technical solution.

Footnotes

[1] DeWitt (then still using the name Carl Bryce Seligman) had only just finished his doctoral thesis on quantum gravity (under Julian Schwinger at Harvard: submitted in December, 1949), and was interested in the possibility of a postdoc at ETH.

[2] “Quantum Gravity: Yesterday and Today.” General Relativity and Gravitation 41, 2009: p. 414.

[3] Selected Papers on Quantum Electrodynamics (Dover, 1958).

[4] In the historical literature of general relativity it is known as “the renaissance of GR”—see, e.g., Jean Eisenstaedt’s “The Low Water Mark of General Relativity, 1925–1955” (in D. Howard and J. Stachel, eds., Einstein and the History of General Relativity, Birkhäuser, 1989: 277–292), David Kaiser, “A \psi is just a \psi? Pedagogy, Practice, and the Reconstitution of General Relativity, 1942-1975” (Studies in History and Philosophy of Modern Physics 29, 1998: 321–338), and AB, Roberto Lalli, and Jürgen Renn, “The Reinvention of General Relativity: A Historiographical Framework for Assessing One Hundred Years of Curved Space-time” (Isis 106, 2015: 598–620).

[5] “A Suggestion for Unifying Quantum Theory and Relativity.” Proceedings of the Royal Society London A 165(921), 1938: p. 291. By this stage, Born was thinking of the problem as involving the mixing of quantum principles with the principle of general covariance, rather than general relativity more broadly conceived—this on account of the fact that “gravitation by its order of magnitude is a molar effect and applies only to masses in bulk, not to the ultimate particles.” However, the idea that general relativity might be employed as part of the basic framework of a future theory of elementary particles became popular in later work.

[6] We see the sources of this in Part V in which the general covariance of general relativity was tackled head on in the context of a parameter formalism and the constrained Hamiltonian formulation.

Acknowledgements

This book benefitted greatly from the support of Jürgen Renn, who not only co-organised and funded (via the MPIWG) two highly successful workshops on the early history of quantum gravity (at Pasadena in 2010 and Berlin in 2011), but also helped fund several research projects closely related to this book. We wish to also acknowledge Diana Kormos-Buchwald and Tilman Sauer (now University of Mainz), of the Einstein Papers Project, who were also responsible for the organisation of the Pasadena workshop. The participants in those workshop sessions deserve thanks for their probing analyses of several of the papers that appear in this volume. We would also like to thank Urte Brauckmann of the MPIWG library for her invaluable help in procuring the reproduction rights to the papers. Dean Rickles would also like to thank the Australian Research Council for its support of his historical research via a Future Fellowship: FT130100466.

Dimensions and Domains

1. Quantum Theory Meets Gravitation: First Encounters

Dean Rickles

These earliest papers (taken from 1916–1921) suffer from the fact that quantum theory was still in an embryonic state. Quantum mechanics and quantum field theory would have to wait several years before they were even properly formulated, and so could be compared with general relativity (as subsequent parts of this volume will cover). Still, already in these early papers, questions were being posed concerning the proper way to understand the relationship between quantum theory and general relativity. In addition to areas in which there might be modifications brought about by one theory to the other (conflict), or overlaps of some kind (coexistence), there were also “borders” raised between the theories that persisted throughout future changes in physics in the period before 1950. Hence, although one cannot reasonably call this research “quantum gravity,” many elements of this pre-history nonetheless directly inspire that later research and, more importantly, serve to demarcate the domains of the ingredient theories.

What is crucial to note in this period is that the developments in quantum theory and in general relativity are often made by the same people, laying the foundations of both simultaneously. For example, as the field equations of general relativity were entering their ultimate form (“the final stage in which the battle over the field equations is being fought out”[1]), Einstein was corresponding with Arnold Sommerfeld about the potential impact of his new theory on contemporary issues in quantum theory. Sommerfeld appears to have thought that general relativity might be of relevance (imposing new constraints and so on) with regard to spectral physics and the Stark effect (Sommerfeld 2000, 438). Einstein writes back:

General relativity is unlikely to be able to assist you, because it practically coincides with the more restricted theory of relativity for those problems. [...] [A]ny other theory that corresponds with relativity in the restricted sense can be taken over in the general theory of relativity through simple transformation, without the latter delivering any new criteria. Thus you see that I cannot help you in the least. [Einstein, letter to Sommerfeld, [Berlin] 9 December 1915 (Schulmann et al. 1998, 159)]

However, in his paper “Näherungsweise Integration der Feldgleichungen der Gravitation” (1916, Chapter 2 in this volume), Einstein claims that his newly constructed theory of gravitation would itself need to be modified by quantum theory along the same lines as classical electrodynamics so as to keep gravitationally radiating systems stable. Hence, without a quantized emission of gravitational energy, the system would (eventually) continuously radiate away all its energy and collapse. It is, perhaps, no surprise that Einstein was working in tandem on the quantum theory of emission and absorption of radiation at the same time he wrote this paper on gravitational radiation, and the same terminology and concepts can be found in both contexts.

The Einstein paper features further, and deeper, analogies between electromagnetism and general relativity, and these analogies continued to play a role in later research on quantum (and classical) gravity.[2] For example, just as an accelerated electrically charged particle will emit electromagnetic waves (at the velocity of light), gravitational waves were a direct prediction of the linear theory and were also transverse and propagated at the speed of light. The linear approach (a first approximation to the full non-linear theory, again analogous to electrodynamics’ standard perturbative approximation where the coupling constant is taken to be small) involves decomposing the metric g_{\mu \nu } apart as follows:

(1.1)

The small perturbations (representing deviations in the spacetime metric away from flat space[3]) \gamma _{\mu \nu } are defined against a flat, Minkowski spacetime background (a “Galilean space” in which x_{4} = it). The calculation follows exactly the same route as for the retarded electromagnetic potentials, giving the various analogous properties mentioned above, including the existence of gravitational waves.

This perturbative approach (employing a background metric against which the properties of the perturbations are defined) forms one of the standard methods in later work. As we see in Part III, Rosenfeld uses just such a framework in his early quantization and it informs the work of Fierz and Pauli in unpacking the properties of gravitational quanta (massless particles with two helicity states)—Jacques Solomon raised an early warning about the problems faced when ignoring the full non-linear theory; Dmitri Ivanenko noted that during a discussion at the Turin Congress (in 1956), Pauli expressed skepticism about the generalisation of results from the weak field approximation to the full theory in the case of gravitation (Ivanenko 1956, 355).

Einstein published a second paper on gravitational waves in 1918, correcting some problems in the 1916 paper and drawing attention to some crucial differences between the electromagnetic and gravitational cases (which had led Einstein to a serious error of calculation).[4] The most important of these concerned the differences in moments of the radiation: dipole in the case of electrodynamics (as he had also assumed held for gravity in the 1916 paper), but quadrupole in the case of gravitation: “[A] mechanical system which permanently retains spherical symmetry cannot radiate” (Einstein 1918, 23). Einstein reiterates the point regarding the likely modifications of general relativity brought about by a completed quantum theory, since the results still indicated energy loss due to thermal agitation (Einstein 1918, 23).

Just one year later Einstein can be found speculating about the role of gravitational fields in the constitution of elementary particles of matter (electrons) in his paper “Do Gravitational Fields Play an Important Role in the Constitution of the Elementary Particles?”.[5] The idea is to get discrete particles (electrons) out of a continuous field theory. John Wheeler would later resurrect something similar in the early 1950s with his “Geon project” (where a geon is a “gravitational electromagnetic entity”), in which the elementary particles are reduced to geometrical and topological aspects of such entities (see, e.g. Wheeler (1955)).[6] Once this shift in Einstein’s thinking had occurred, he didn’t waver again and sought only classical theories as opposed to quantum gravity theories. However, there were additional important developments in Einstein’s work. Not least among these was the result, obtained with Infeld and Hoffmann (Einstein, Infeld, and Hoffmann 1938), that enabled particle trajectories (equations of motion) to be derived from the field equations—this would have a direct impact on Peter Bergmann’s earliest work on quantum gravity (Bergmann 1949). Bergmann was concerned with using this feature of generally covariant theories to eliminate the divergences caused by particle interactions (infinite interaction terms) in field theories: field singularities (particles) can be determined from the field equations. In this project, it is clear that Bergmann has Einstein’s old question, of the constitution of elementary particles, in mind.

The general idea that quantum and gravity would have to meet in some way in a complete formulation of the world seems to have been accepted by other physicists of the time, and motivated subsequent early work and commentaries on the subject. While there were many remarks pointing to the clash, there wasn’t much real detailed work to resolve the problem, or even any systematic investigation, until Léon Rosenfeld’s efforts in 1930 (following on from initial work on quantum electrodynamics), discussed in the next parts. As mentioned above, until quantum theory was established on a firmer footing, it would have been impossible to tackle the problems of unification.

While the majority of the Eddington paper (Chapter 3), from 1918, amounts to a popular presentation of general relativity[7], it also contains the first statement that we know of concerning the relevance of the Planck length in connection with (quantum) gravitational physics, and includes a statement of its importance in any future theories that wish to merge gravitation ( cG) and quantum theory ( h).[8] Eddington sees that the minute nature of the fundamental length generated from the theoretical constants meant that the structure of a unifying theory will be hidden until we can probe down to the “quadrillionth or quintillionth of a centimetre”.[9] Though the numerical value he cites is not as we write it today (perhaps a computational error?), given that he speaks of the fundamental unit length one achieves by combining the three basic universal constants cGh, and there is only one way to do this, we must assume that he has in mind the Planck length nonetheless.

Hence, by 1918 we already have two arguments (however coarsely and briefly expressed) for the necessity of a unification of quantum theory and general relativity (or gravitation and quantum phenomena), both for consistency and for reasons of theoretical unity. It is not yet quantum gravity in the sense of quantization that is being proposed. Indeed, nothing is being proposed at this stage; rather, it is left as a future project. Both Einstein and Eddington would soon diverge rather radically from the three motivations outlined earlier. Though Eddington stuck to the project of unification of quantum theory and general relativity, until his death, his approach departed from the reductionist (deeper probing) method he suggests in Paper 2. Perhaps forced by the sheer distance of scales (from known physics) we see in the Planck units, Eddington began to employ a non-experimental methodology, culminating in his (posthumously published) Fundamental Theory (Cambridge University Press, 1946).

George Barker Jeffery (9 May 1891—27 April 1957) was one of the few physicists to directly acknowledge Einstein’s remarks about the probable modifications that quantum theory would bring about in the theory of gravitational radiation.[10] The main result of the paper (Chapter 4) is not entirely novel, though he appears to have discovered them independently: Jeffery himself notes that he had been unaware until publication that Nordström had only just derived similar results to his. The central outcome is an extension of Schwarzschild’s line element (of space surrounding a point) to the combined electromagnetic and gravitational case (considering particles with charge and inertial mass), done by modifying the \gamma-term as:

(1.2)

In this case the point is a singularity of both fields (i.e. there are two singularities), rather than the gravitation field alone:

(1.3)

Oliver Lodge discusses Jeffery’s proposal as an attempt to “ascertain something about the state of the aether close to an electron” (Lodge 1921, 392), which indicates something about the transitionary state of physics at the time. But he also rightly points out that the paper involves thinking about the status of fields near to points, and the question of whether the elementary particles really have structure, which itself leads to the additional question of radiation’s interaction with elementary particles in such cases (especially as regards absorption of radiation by elementary particles). As he puts it: “a study of what happens to radiation when it impinges on, or penetrates between the ultimate elements of matter—in fact, a study of the whole behaviour of a stream of radiation at its two ends, the source and the sink—is obviously of great importance” (Lodge 1921, 392). Such reasoning is hardly shot down by the act of bringing those “ultimate elements of matter” in line with quantum principles.

The ultimate aim of Jeffery’s paper was to show how the gravitational field might be involved in the structure of the electron (in line with Einstein’s “turning point” mentioned above), with the conclusion that the electrical and mass potentials would offer some kind of stabilizing effect by opposing each other.[11] This brings him to Einstein’s rather pessimistic remarks about the fate of general relativity at the hands of quantum theory from the 1916 paper (Chapter 2), regarding the instability of the atom in the face of continuous classical gravitational radiation. Jeffery believed that the problem could be evaded. Einstein was not convinced by Jeffery’s idea:

I unfortunately cannot share your optimism regarding the solution to the quantum problem. I believe that the theory of relativity does not bring us a step closer, at least in its current form. I am convinced that the two-body problem will not lead to a discrete manifold of paths but to a continuous one. (Einstein, letter to Jeffery [Berlin], 18 March 1921 (Kormos-Buchwald et al. 2009, 85))

But, as we have seen, Einstein had already started down a path that followed the spirit if not the letter of Jeffery’s approach (namely, using GR, or some modification of it, to recover quantum behaviour). In other words, Einstein’s earlier assumption about quantum restrictions of gravitation were replaced by the view that quantum phenomena are to be derived from general relativity and thus are not fundamental. However, here Einstein does not indicate this aspect of his thinking to Jeffery, and indeed the quote above looks largely negative as far as the entire project of getting quantum from relativity goes.

Though Einstein was followed by several others along the “unified field theory” path, the majority view was that his earlier pessimism should and could be responded to without rejecting quantum theory, as we see in the papers that follow. In the next part, the two ingredient theories are worked out in more detail along exactly the lines of figuring out how generally relativistic principles and quantum principles can co-exist, at least in a formal-structural sense.

References

Bergmann, Peter G. (1949). “Non-Linear Field Theories.” Physical Review 75 (4): 680–85.

Eddington, Arthur S. (1922). “The Propagation of Gravitation Waves.” Proceedings of the Royal Society of London A 102 (716): 268–82.

Einstein, Albert (1918). “Über Gravitationswellen.” Sitzungsberichte Der Königlich Preußischen Akademie Der Wissenschaften Berlin, 154–67.

Einstein, Albert, Leopold Infeld, and Banesh Hoffmann (1938). “The Gravitational Equations and the Problem of Motion.” Annals of Mathematics 39 (1): 65–100.

Gorelik, Gennady E. (1992). “First Steps of Quantum Gravity and the Planck Values.” In Studies in the History of General Relativity, edited by Jean Eisenstaedt and A. J. Kox. Boston: Birkhäuser.

Ivanenko, Dmitri (1956). “Non-Linear Generalisations of the Field Theory and the Constant of Minimal Length.” Supplemento Al Nuovo Cimento 6: 349–55.

Kennefick, Daniel (1998). “History of the Radiation Reaction Problem.” In The Expanding Worlds of General Relativity, edited by Hubert Goenner, Jürgen Renn, Jim Ritter, and Tilman Sauer. Boston: Birkhäuser.

Kormos-Buchwald, Diana L., Ze’ev Rosenkranz, Tilman Sauer, József Illy, and Virginia I. Holmes (2009). The Collected Papers of Albert Einstein, Vol. 12. Princeton: Princeton University Press.

Lodge, Oliver (1921). “The Gravitational Field of an Electron.” Nature 2691 (107): 392.

Schulmann, Robert, A.J. Kox, Michel Janssen, and József Illy (1998). The Collected Papers of Albert Einstein, Volume 8. Princeton: Princeton University Press.

Sommerfeld, Arnold (2000). Wissenschaftlicher Briefwechsel. Band 1: 1892–1918. München: GNT-Verlag.

Vizgin, Vladimir P. (2011). Unified Field Theories: In the First Third of the 20th Century (Translated by Julian Barbour). Boston: Birkhäuser.

Wheeler, John A. (1955). “Geons.” Physical Review 97 (2): 511–36.

Footnotes

[1] As described in a letter to Sommerfeld, [Berlin] 9 December 1915 (Schulmann et al. 1998, 159).

[2] Gennady Gorelik (1992) briefly discusses the role of the electrodynamical-gravitational analogy in this phase of Einstein’s thinking. He argues, rightly we think, that an overly strong analogy persisted in the decades following, fooling physicists into thinking that the task of constructing a quantum theory of gravity was going to be along the same lines as the quantum treatment of electrodynamics. Much of this was an artefact of the linearised approximation scheme Einstein used in 1916, which eradicates the distinguishing (and very difficult) features of gravitation. Given the tractability of the linear approximation, this was the scheme that was used in most of the early direct quantization attempts.

[3] A “deviation of the continuum from one that is field-free” in Einstein’s own words (Gorelik 1992).

[4] The prediction of waves would form one of the central motivations for quantum gravity in the 50s. Readers will no doubt be aware of Einstein’s double about-face in 1936, leaving him back where he started! Einstein was briefly fooled into thinking that the existence of gravitational waves was an artefact of the linear theory (true only for weak fields), so that when the non-linear field equations were used instead, the waves were no longer physical or energy-transporting. Together with Nathan Rosen, he had originally argued that (cylindrical) gravitational waves are unreal, in a paper entitled “Are Gravitational Waves Real?” (submitted to Physical Review). But following Howard Robertson’s referee report pointing out a serious flaw in their paper (initially dismissed as erroneous by Einstein), he switched his position (though publishing in a different venue, the Journal of the Franklin Institute, and without acknowledgement of Robertson’s helpful critique). For a discussion of this episode, see Kennefick (1998)—the title of Kennefick’s book (Travelling at the Speed of Thought) refers to Eddington’s characterisation of gravitational waves in 1922 (Eddington 1922, 269).Of course, as a practical prediction the result suffered from the fact that there were no known sources (terrestrial or astrophysical) that could generate energies of sufficient magnitude to be detected—this changed in the 1960s.

[5] According to Vizgin (2011, 163), this marks the birth of Einstein’s switch to “unified field theory,” in which gravitational (and electromagnetic) fields are fundamental, and discrete atomic particles emergent features. The basic question of whether and to what extent the gravitational field might be involved in the structure of the electron also plays a part in Paper 3, by Jeffery (see below).

[6] In order to recover such things as charge and spin, the geon project became rather complex, with e.g. wormholes (multiply connected space) being used to recover charge phenomenology. For more on the history, see Dieter Brill (John Wheeler’s student): http://mediathek.mpiwg-berlin.mpg.de/mediathekPublic/versionEins/Conferences-Workshops/Quantum-Gravity/Thursday/J-A-Wheeler-Geons.html. The geon was originally called “kugelblitz” (ball lightning) by Wheeler. It is interesting to speculate on the origins of the concept in his work on plasma physics for Project Matterhorn: one of his geon models was a toroidal geon of the kind that is often invoked in ball lightning research (where it is modeled as a stable, spinning plasma toroid).

[7] This underplays other important aspects of the paper. For example, it includes a particular perspective on the physical content of the general theory of relativity involving the idea that it is a theory of space considered as “the scaffolding constructed from our measures”—a view that retains a degree of modernity.

[8] Hence, here we disagree with Gorelik who assigns priority to Matvey Bronstein in 1936 (see Gorelik (1992, 367)). However, it is true that the first detailed analysis of the extent to which the Planck values encode information about the quantum limits of general relativity (in the sense of measurability analysis) was due to Bronstein. Note that the issue of domains of applicability of theories would return again, firstly in the context of the debate about measurability limitations (alluded to already), and also in the context of a minimum length implied by quantum theory which are also linked to the former in some cases.

[9] Recall that in England a quintillionth is 10^{-30}. For two more decades, the smallest length relevant to physics was set to 10^{-13} in line with the scales set by the existence of known particles (rather than any future combination of relativity and the quantum).

[10] Like Eddington, Jeffery was a Quaker, and had spent time in prison in 1916 as a conscientious objector—Einstein was aware of this writing wittily in his letter to Jeffery that it “is a highly welcome fact that a considerable portion of England’s learned world upholds the pacifist ideal” (Kormos-Buchwald et al. 2009, 85).

[11] This, he thought, might be the result of an analysis of the two body problem in his scheme: a subject for future work. Of course, this problem (exact solutions for the gravitational two-body problem for two point singularities) would ultimately have to wait for new methods in the initial value problem and numerical techniques—though the Einstein, Hoffmann, Infeld paper (1938) is precisely along these lines, and, as we have seen, Bergmann used this method precisely for its ability to avoid interaction divergences. Curiously, Jeffery argues that without including electric charge, the point singularity blows up so that one has “not a solution with a single point singularity, but a solution with a point singularity surrounded by a spherical surface of singularity” (see p. 42). Adding charge eliminates all zeros and infinities other than that occurring at r=0. Even more curiously, Jeffery speculates that given the light deflection features of general relativity “it would seem to be not impossible that a ray which passed sufficiently close to an attracting particle might be so strongly deflected that it would be permanently entrapped by the particle” (see p. 42). Again, the introduction of charge is invoked to prevent such light sinks (it seems clear that something like a black hole is being suggested here).

2. Albert Einstein (1916): Näherungsweise Integration der Feldgleichungen der Gravitation

Albert Einstein (1916). Näherungsweise Integration der Feldgleichungen der Gravitation. Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte.

3. Arthur Stanley Eddington (1918): Gravitation and the Principle of Relativity II

Arthur Stanley Eddington (1918). Gravitation and the Principle of Relativity II. Nature 101(2524): 34–36.

4. George B. Jeffery (1921): The Field of an Electron on Einstein’s Theory of Gravitation

George B. Jeffery (1921). The Field of an Electron on Einstein’s

Theory of Gravitation. Proceedings of the Royal Society of London. Series A 99 (697), 123–134.

Shifting Boundaries between Quantum Theory and Relativity

5. Where to Start? First Interactions between Wave Mechanics and General Relativity

Alexander Blum

In the years 1925 to 1927, the old quantum theory was replaced by the newly developed theory of quantum mechanics, which grew out of the matrix mechanics of Heisenberg, Born, and Jordan and the wave mechanics of Schrödinger. Both of these theories were initially formulated entirely non-relativistically. But it was clear from the outset that contact would have to be made with the special theory of relativity for two important reasons: On the one hand, the mechanics would have to be complemented with a quantum electrodynamics (QED), in order to describe the emission and absorption of radiation, as well as the particulate properties of light itself, which by this time (in the wake of the discovery and interpretation of the Compton effect) was a generally accepted fact. On the other hand, the mechanics itself would have to be made relativistic, as it was known, already since the mid 1910s, that relativistic corrections to the kinematics of the electron would have a measurable effect in the fine structure of atomic spectra.

It was Schrödinger’s wave mechanics, rather than matrix mechanics, that provided the ideal starting point for a relativistic kinematics of matter: One needed to find a new, relativistic matter wave equation, but at first glance there were no immediate other conceptual difficulties, such as the problem of a non-commuting time variable in matrix mechanics. Schrödinger himself had initially attempted to find a relativistic wave equation, following de Broglie’s program of matter waves, which had been formulated in a relativistic manner. But de Broglie had stopped short of addressing the dynamical equations. Schrödinger in fact arrived at the Klein Gordon equation, but he dismissed it, due to its empirical inadequacy (later understood as the absence of spin in the Klein Gordon equation). Others were not as scrupulous, and the Klein Gordon equation was rediscovered (and published) multiple times in the immediate aftermath of Schrödinger’s first papers on wave mechanics.[1]

Which role did general relativity play in this context? In full analogy with the case of special relativity, two distinct problems can be distinguished. On the one hand, there is the question of a quantum theory of the gravitational field. This will be discussed in the next part. On the other hand, there is the question of how quantum matter interacts with a gravitational field. We will be discussing this aspect first, because it is primary both logically and historically: Many of those who attempted a relativistic generalization of Schrödinger’s program didn’t see a reason to stop at special relativity—after all, the final quantum wave equation should also be compatible with general relativity and be able to describe the interaction of microscopic matter with a gravitational field. In two cases, the re-discovery of the Klein Gordon equation thus actually occurred in the context of curved space-time, in the work of Théophile de Donder and Oskar Klein.

It should be noted that at the time, quantum mechanics was far from being well enough established for these works to be viewed as merely an application of the new quantum theory to general relativity. Rather, by incorporating wave mechanics within the better-established framework of general relativity, they sought to put the former on a surer foundation. From a modern viewpoint, De Donder’s papers are notable for delivering the first construction of a Klein Gordon equation in curved spacetime (Chapter 7). But for de Donder this was not the central point. Rather, the generalization of Schrödingers “quantization” procedure for arriving at the Schrödinger equation to the generally relativistic case was in some sense meant to motivate the quantization procedure itself, even if de Donder’s reasoning on this point is rather sketchy (Chapter 6).

In a similar vein, Klein’s attempt to bring together wave mechanics with the five-dimensional extension of GR, proposed by Kaluza (1921) in order to unify gravity and electromagnetism, might simply be read as the construction of a Klein Gordon wave equation in curved spacetime with an electromagnetic potential. But Klein’s main ambition was to see whether the interpretation of Schrödinger’s waves as waves in actual spacetime, which the likes of Bohr and Heisenberg doubted from the very start, might in fact be saved in a five-dimensional spacetime (Chapter 8). These hopes, cautiously voiced by Klein in the conclusion to his paper, were soon to be dashed, when Klein realized that such a theory would never be able to explain Planck’s law of black body radiation, the problem that had led to quantum theory in the first place. In 1927, Klein converted to the probabilistic camp, now placing his hopes for a spatio-temporal description in the newly emerging quantum field theory.

This did not immediately change Klein’s research program: He continued to elaborate his classical five-dimensional field theory, but now viewed it merely as a starting point for quantization. While work on a full theory of quantum electrodynamics was well underway at this time, the quantization of gravity was not viewed as an immediate concern. Klein’s second paper (Chapter 9) thus begins with some remarks on why general relativity would need to be modified according to the postulates of quantum theory. This is the first published argument for the necessity of quantizing the gravitational field. He further argued that gravity and electrodynamics should be quantized simultaneously, that is, that one should quantize a unified field theory rather than simple Maxwellian electrodynamics (interacting with a Klein Gordon matter field), as this would, in particular, allow for a unification of the conservation laws of energy-momentum and of charge.

Indeed, if Klein’s later recollections are to be believed, his five-dimensional field theory initially formed the classical basis for Heisenberg’s and Pauli’s attempts at formulating QED.[2] That is, until the Dirac equation came along in 1928 and ousted the Klein Gordon equation as the best description of relativistic quantum matter. Initially, Dirac saw one main advantage in his new equation over the old Klein Gordon one, namely that it could be interpreted as a one-particle, quantum mechanical Schrödinger equation (Dirac 1928). But even those who believed that any theory of matter waves would necessarily have to be second-quantized immediately realized the superiority of the Dirac equation, incorporating as it did the spin of the electron (and the proton).

In the wake of the Dirac equation, work immediately began on its integration into general relativity, the most immediate goal being the formulation of the Dirac equation in a curved (but non-dynamical) space-time. The first step in this direction was taken by Hugo Tetrode, already in 1928 (Chapter 10). The general idea was the following: In Dirac’s derivation, the flat Minkowski space-time metric \eta _{\mu \nu } shows up in the algebra of the Dirac \gamma matrices:

(5.1)

Tetrode’s starting point was now to replace the Minkowski metric in this relation with a generalized non-flat metric g_{\mu \nu }, turning the (elements of the) \gamma matrices from constants into space-time dependent quantities, determined by the value of the metric at a given space-time point. But this determination was not unique, in fact it did not even offer a clear prescription for constructing the \gamma matrices when the metric was given. Tetrode only presented some tentative attempts at finding such a prescription, but, soon after, Eugene Wigner proposed that one could answer this question by making use of a formalism that Einstein had developed in that same year (1928), but in an entirely different context (Wigner 1929).

Einstein’s idea was to replace the metric g_{\mu \nu } as the fundamental quantity of his general theory of relativity with so-called “vierbeine” (tetrads), that is, for each point in spacetime a local coordinate system spanned by four orthonormal basis vectors h^{\nu }_a, with a running from one to four (Einstein 1928). The metric at a given point was determined by the tetrad to be

(5.2)

but the metric by itself did not determine the tetrads. Indeed, the metric is unchanged by an arbitrary local Lorentz transformation of the tetrads

(5.3)

Einstein now proposed that the relative orientation of the tetrads was not redundant information, but actually carried physical meaning. In other words, once one has defined an x axis at some point in spacetime, it makes a difference what I call x axis in some other point—the only allowed transformations are then global Lorentz transformations, which conserve the relative orientation of the tetrads. This allowed one to speak of vectors at different spacetime points as parallel if they had the same components with regard to their local tetrad; hence Einstein’s theory went by the name of teleparallelism. Through the notion of teleparallelism, the tetrads thus defined a second connection (beside the usual Levi-Civita connection one could obtain from the metric tensor alone) which could be used to construct new tensors and invariants, which could in turn be used to write down more general field equations. Einstein believed that he could thereby construct combined field equations for the gravitational and electromagnetic fields.

Wigner realized that, independent of these unified field theory considerations,[3] given local tetrads one could give a unique prescription for the construction of the \gamma matrices in curved spacetime as

(5.4)

where the \tilde{\gamma } are the usual (constant) Dirac matrices; the Dirac matrices on curved spacetime \gamma can then easily be shown to obey the correct anti-commutation relations. The open question was then: Given the Dirac matrices in curved spacetime, how were they to be incorporated into the Dirac equation? Both Tetrode and Wigner had tried this, but they had mainly concerned themselves with the question of how space-time dependent \gamma matrices were to be integrated into the Dirac equation, which of course contained derivatives with respect to space and time. The general covariance of the wave equations they ended up with was, however, very questionable. This question was soon taken up by Vladimir Fock, initially in collaboration with Dimitri Ivanenko.

Fock’s main innovation was introducing the notion of the covariant derivative of a spinor, which allowed for a manifestly covariant Dirac equation in curved spacetime (Chapter 11). The connection appearing in the covariant spinor derivative was determined by the Dirac matrices in curved spacetime, which in turn were determined by the tetrads. But Fock could show that his Dirac equation was indeed covariant under local Lorentz transformations, that is, that the choice of tetrads (beyond what was determined by the metric) played no physical role. The tetrads were thus demoted to mere mathematical tools, in contrast with the physical role they had played in Einstein’s teleparallelism.

A very similar approach was developed independently by Hermann Weyl (Chapter 12).[4] But Weyl’s approach differed fundamentally from Fock’s in its perceived goal. Fock came from quantum theory and viewed his work as an extension of Dirac’s theory to the case of curved spacetime. Weyl, on the other hand, is a representative of the viewpoint, which we already encountered in the work of de Donder and the early Klein, that general relativity might help in solving foundational issues in quantum mechanics. For while the original problems of motivating the quantization procedure or interpreting the wave function had largely been set aside by 1929 (the year of Weyl’s and Fock’s work), the Dirac equation brought with it new fundamental difficulties, most notably the problem of the negative energy states.

Weyl attempted to solve this difficulty by replacing the four-spinors of Dirac with the two-spinors that now carry Weyl’s name. This removed the negative energy solutions, but at the same time also prevented the inclusion of the usual explicit mass term in the wave equation. Weyl expressed the rather vague hope that the mass might arise through the coupling to a gravitational field and, to this end, constructed an essentially classical field theory of coupled spinor, gravitational and electromagnetic fields, which, just like Klein’s five-dimensional field theory two years earlier, was eventually supposed to be quantized. In order to couple the spinors to the gravitational field, he introduced the same tetrad-based covariant spinor derivative that Fock had constructed. For the coupling of the spinors to the electromagnetic field, he discovered, almost in passing, the local gauge invariance of the Dirac equation, which he not only related to, but in fact tried to deduce from local Lorentz invariance.

The work of Weyl and Fock was received in two distinct ways: Some, coming to the problem from general relativity, attempted to further pursue the solution of the negative energy problem along these lines, most notably Jan Schouten, who attempted to show how a mass term for Weyl spinors might arise in five- instead of four-dimensional curved spacetime (Schouten 1931). In the quantum community, on the other hand, the work of Weyl and Fock was viewed as the classical basis for a relativistic quantum field theory of gravitation, electromagnetism and charged Dirac particles, and an attempt to quantize it was undertaken almost immediately by Léon Rosenfeld (see the following part of this book). It was important for this undertaking that Weyl and Fock had not only shown how to couple spinor waves to a gravitational field, but had also identified the degrees of freedom in general relativity that were supposed to be subjected to quantization in such a coupled theory, the tetrads.

In 1932, Schrödinger showed that this was not the unique way of integrating spinors into general relativity. He took the spacetime dependent Dirac matrices as fundamental degrees of freedom, instead of the tetrads (Chapter 13). This brought the question of which basis to choose for the Dirac matrices back onto the table: In the tetrad prescription, equation 5.4, one simply had to initially pick some basis for the constant Dirac matrices; everything else then followed from the tetrads and their transformation properties. But if the \gamma matrices themselves were supposed to be the dynamical variables, no direct reference could be made to the constant matrices. Schrödinger thus imposed further hermiticity conditions, which defined the space-time dependent Dirac matrices, given a metric, up to a unitary transformation in spin space. These unitary transformations were not only, as Schrödinger put it, “benign and irrelevant,” they also corresponded exactly to the local Lorentz transformations in the Weyl-Fock tetrad formalism. What had thus, using tetrads, looked like the rotation of local coordinate axes, in Schrödinger’s formalism was interpreted as a (local) transformation in spin space. This latter point, in particular, was soon after further worked out by Bargmann (1932) and Pauli (1933).

A further alternative was worked out by Bartel van der Waerden and Leopold Infeld (Chapter 14), based on a generalization of van der Waerden’s special relativistic spinor calculus (Waerden 1929).[5] They defined, instead of the local coordinate basis of the tetrad formalism, local spinor spaces. The separation between local spin space transformations and coordinate transformations was thus now taken as the starting point, rather than as a result of the analysis. As in special relativistic spinor calculus, where vectors are constructed from spinors through the Pauli matrices, the connection between spinor space and spacetime was established through a set of local, spacetime dependent generalized Pauli matrices. These matrices replaced the metric as fundamental degrees of freedom. They were thus the two-spinor analog of Schrödinger’s spacetime dependent Dirac matrices. In a sense, Infeld/van der Waerden was to Schrödinger as Weyl was to Fock: The same general idea, but with two-spinors instead of four-spinors. Van der Waerden viewed his formalism as a considerable simplification of Schrödinger’s four-dimensional framework, since one no longer needed to employ hermiticity conditions in order to fix the dynamical matrix degrees of freedom.[6]

Both the Schrödinger and the Infeld-van der Waerden formalism had to wait quite a while until they were put to use. Even though Pauli immediately suggested to Rosenfeld that he should repeat his analysis of 1930, this time quantizing the Dirac matrices instead of tetrads,[7] Schrödinger’s reformulation was, to the best of my knowledge, not used as the basis for a quantization of gravity until Bryce de Witt’s thesis work in the late 1940s. And, still in 1951,[8] Schrödinger remarked to Fredrik Belinfante: “I do not set any store in my paper of 1932 on Dirac’s theory in a general metric. It was at the time a smart exercise, but it came to nothing.” Van der Waerden was equally dismissive of his work. His formalism had to wait even longer, until the late 1950s, before it was widely received, mainly, however, in the context of mathematical relativity, rather than as a basis for quantization.[9]

In any case, by the early 1930s it appeared that the classical field theory of relativistic quantum matter interacting with the gravitational field had been worked out, in several closely related mathematical formulations. The core difficulties were seen to lie in the quantization procedure, which, already for the simpler case of electrodynamic interactions, led to uninterpretable infinities in the higher orders of perturbation theory.

In 1938, however, a study by Einstein, Infeld, and Banesh Hoffmann (EIH) appeared, which appeared to indicate that there were serious advantages to starting with a classical theory of matter as point particles, in contradiction, ironically, with Einstein’s program of a unified field theory (Chapter 15). EIH could show that if one treated matter particles as singularities in the gravitational field , one could (in stepwise approximations) derive the equations of motion for these singularities from the field equations alone, without having to additionally postulate the geodesic equations for particle motion. The prospect of carrying this feature of general relativity over to the quantum theory made a classical theory of point particles (rather than a classical theory of matter waves to be second-quantized) an attractive starting point for a quantum theory of gravitation. This approach was taken by Peter Bergmann, as will be discussed in the final part of this book.

Several different possible starting points for a quantization of gravity, that is, a modification of general relativity according to the principles of quantum theory, had thus been set. But the inverse program, the modification of quantum theory following considerations from general relativity, was not dead. As we have seen, this program by the late 1920s mainly centered around the idea that the inclusion of general relativity might somehow determine the mass of the elementary particles, which were after all purely empirical parameters in the wave equations of quantum mechanics and field theory.

A variant of this idea, which brought in ideas from relativistic cosmology, had been formulated early on by Arthur Eddington (1923). He hypothesized that the radius of the electron (proportional to the inverse mass) should be related (in fact proportional) to the curvature radius R of the universe as a whole, still viewed at the time as static and spherical (an Einstein universe). Eddington further elaborated on this idea 13 years later in the context of his notorious and oft-ridiculed relativistic theory of protons and electrons (Eddington 1936),[10] obtaining the necessarily huge proportionality constant through the square root of the number N of electrons (or protons) in the entire universe.

While this work was almost universally regarded as obscure and incomprehensible, some of the general ideas concerning the relation between cosmology and elementary particles were taken up by Erwin Schrödinger.[11] Schrödinger saw some promise in Eddington’s new work, inasmuch as it was a specification of the original idea of connecting the electron mass with cosmology, now also incorporating the tools of wave mechanics. In particular, Eddington had derived the rest mass energy of an elementary particle as the energy of the first excited state for the system of all the identical fermions in the universe. The curvature radius of the universe entered through the boundary conditions for the wave equation, the total number of particles in the universe through the fact that one was looking at energies near the Fermi surface. In a stripped-down version, which Schrödinger presented at the Galvani Bicentennial in 1937 (Schrödinger 1938b) and which made no more reference to the more eccentric aspects of Eddington’s theory (such as the attempt to prove that the fine structure constant was necessarily the reciprocal of an integer),[12] the central relation read:

(5.5)

where the dots refer to undetermined constants. In order to obtain such a relation from the accepted wave mechanics, rather than from Eddington’s novel theory, Schrödinger devoted considerable energy to the explicit construction of the solutions of the Maxwell and Dirac equations in a spherical universe (Schrödinger 1938a), also studying the non-static, expanding case, which was by that time heavily favored (Schrödinger 1940). But after three years of working on this subject, he had to conclude in a small note that one would, starting from very general assumptions, always obtain the cubic root of N in Eddington’s relation, instead of the square root, and thus obtain much too small masses for the elementary particles (Chapter 16).

The idea that general relativity, in particular relativistic cosmology, might directly determine the microscopic quantum wave equations thus remained nothing more than a tantalizing possibility. In contrast, the program of quantizing the gravitational field, while technically challenging, seemed a fairly straightforward task. In this section, we have discussed the necessary groundwork for this task, that is, the establishment of classical theories of matter interacting with the gravitational field that went beyond the very unspecific energy-momentum tensor of matter appearing in the Einstein equations. In the next section, we will discuss the first attempts at an actual quantization of such theories.

References

Bargmann, Valentine (1932). “Bemerkungen Zur Allgemein-Relativistischen Fassung Der Quantentheorie.” Sitzungsberichte Der Preussischen Akademie Der Wissenschaften, 346–54.

Dirac, Paul A. M. (1928). “Zur Quantentheorie Des Elektrons.” In Quantentheorie Und Chemie, edited by Hans Falkenhagen, 85–94. Leipzig: S. Hirzel.

Eddington, Arthur S. (1923). The Mathematical Theory of Relativity. Cambridge: Cambridge University Press.

——— (1936). Relativity Theory of Protons and Electrons. London: Cambridge University Press.

Einstein, Albert (1928). “Riemann-Geometrie Mit Aufrechterhaltung Des Begriffes Des Fernparallelismus.” Sitzungsberichte Der Preussischen Akademie Der Wissenschaften, 217–21.

Goenner, Hubert F. M. (2004). “On the History of Unified Field Theories.” Living Reviews in Relativity 7 (2): 1–153.

Kaluza, Theodor (1921). “Zum Unitätsproblem Der Physik.” Sitzungsberichte Der Preussischen Akademie Der Wissenschaften, 966–72.

Kilmister, Clive W. (1994). Eddington’s Search for a Fundamental Theory: A Key to the Universe. Cambridge: Cambridge University Press.

Kragh, Helge (1984). “Equation with the Many Fathers: The Klein-Gordon Equation in 1926.” American Journal of Physics 52 (11): 1024–33.

——— (2015). “On Arthur Eddington’s Theory of Everything.” arXiv:1510.04046.

Meyenn, Karl von, ed. (1993). Wolfgang Pauli: Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.a. Vols. Vol. 3: 1940–1949. Berlin: Springer.

Pauli, Wolfgang (1933). “Über Die Formulierung Der Naturgesetze Mit Fünf Homogenen Koordinaten. II. Die Diracschen Gleichungen Für Die Materiewellen.” Annalen Der Physik 5 (18): 337–72.

Penrose, Roger (1960). “A Spinor Approach to General Relativity.” Annals of Physics 10 (2): 171–201.

Schneider, Martina R. (2010). “Die Physikalischen Arbeiten Des Jungen B. L. van Der Waerden.” PhD thesis, Bergische Universität Wuppertal.

Scholz, Erhard (2005). “Local Spinor Structures in V. Fock’s and H. Weyl’s Work on the Dirac Equation (1929).” In Géométrie Au Vigntième Siècle, 1930–2000, edited by D. Flament, J. Kouneiher, P. Nabonnand, and J.-J. Szsceciniarz, 284–301. Paris: Hermann.

Schouten, Jan A. (1931). “Dirac Equations in General Relativity.” Journal of Mathematics and Physics 10: 239–83.

Schrödinger, Erwin (1938a). “Eigenschwingungen Des Sphärischen Raumes.” Commentationes Pontificiae Academiae Scientiarum 2: 321–64.

——— (1938b). “Sur La Théorie Du Monde d’Eddington.” Il Nuovo Cimento 15: 246–54.

——— (1940). “Maxwell’s and Dirac’s Equations in the Expanding Universe.” Proceedings of the Royal Irish Academy 46A: 25–47.

Urbantke, Helmuth (1992). “Schrödinger and Cosmology.” In Studies in the History of General Relativity, Einstein Studies Vol. 3, edited by Jean Eisenstaedt and A. J. Kox, 453–59. Boston: Birkhäuser.

Waerden, Bartel van der (1929). “Spinoranalyse.” Nachrichten von Der Gesellschaft Der Wissenschaften Zu Göttingen, Math.-Phys. Klasse, 100–109.

Wigner, Eugene P. (1929). “Eine Bemerkung Zu Einsteins Neuer Formulierung Des Allgemeinen Relativitätsprinzips.” Zeitschrift Für Physik 53 (7): 592–96.

Footnotes

[1] For a detailed history of the Klein Gordon equation, see Kragh (1984).

[2] Oral history interview conducted by J. L. Heilbron and L. Rosenfeld on 25 February 1963, Niels Bohr Library and Archives, American Institute of Physics, College Park, MD, USA, http://www.aip.org/history/ohilist/4709_3.html.

[3] As we shall see, this is just one instance of the close connection between developments in unified field theory and the Dirac equation in curved space-time. For a detailed discussion from the viewpoint of the history of unified theory, see Goenner (2004).

[4] For a detailed discussion of Weyl’s, and also Fock’s, work and its context, see Scholz (2005).

[5] For a detailed study of van der Waerden’s work in physics, including this paper, see Schneider (2010).

[6] See the letter from van der Waerden to Schrödinger, 14 June 1932, Schrödinger papers, Vienna University Library.

[7] Letter from Pauli to Rosenfeld, 25 November 1932, reprinted in Meyenn (1993).

[8] Letter from Schrödinger to Belinfante, 16 March 1951, Archive for the History of Quantum Physics.

[9] See, in particular, Penrose (1960).

[10] For a benevolent and insightful reading of this book, see Kilmister (1994). For more on the historical background, see Kragh (2015).

[11] For more on this work in the context of Schrödinger’s work on cosmology more generally, see Urbantke (1992).

[12] Still, reactions were mixed. On 18 February 1938, Sommerfeld wrote to Schrödinger: “It is wonderful that you are now seriously working on Eddingt.[on].” (Es ist wundervoll, dass Sie mit Eddingt.[on] Ernst machen) but also talked about Bohr’s “arrogant dismissal” (bonzenhafte Ablehnung), Schrödinger papers, Vienna University Library.

6. Théophile de Donder and Frans Henri van den Dungen (1926): La quantification déduite de la Gravifique einsteinienne

Théophile de Donder and Frans Henri van den Dungen (1926). La quantification déduite de la Gravifique einsteinienne. Comptes rendus, 183: 22–24.

7. Théophile de Donder (1926): Application de la quantification déduite de la Gravifique einsteinienne

Théophile de Donder (1926). Application de la quantification déduite de la Gravifique einsteinienne. Comptes rendus, 183: 594–595.

8. Oskar Klein (1926): Quantentheorie und fünfdimensionale Relativitätstheorie

Oskar Klein (1926). Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik 37: 895–906.

9. Oskar Klein (1927): Zur fünfdimensionalen Darstellung der Relativitätstheorie

Oskar Klein. Zur fünfdimensionalen Darstellung der Relativitätstheorie. Zeitschrift fur Physik, 46: 188–208.

10. Hugo Tetrode (1928): Allgemein-relativistische Quantentheorie des Elektrons

Hugo Tetrode (1928). Allgemein-relativistische Quantentheorie des Elektrons. Zeitschrift für Physik, 50: 336–346.

11. Vladimir Fock (1929): Geometrisierung der Diracschen Theorie des Elektrons

Vladimir Fock (1929), Geometrisierung der Diracschen Theorie des Elektrons. Zeitschrift für Physik, 57: 261–277.

12. Hermann Weyl (1929): Elektron und Gravitation

Hermann Weyl (1929). Elektron und Gravitation, Zeitschrift für Physik, 56: 330–352.

13. Erwin Schrödinger (1932): Diracsches Elektron im Schwerefeld I

Erwin Schrödinger (1932). Diracsches Elektron im Schwerefeld I. Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 1932, 105–128.

14. Leopold Infeld and Bartel van der Waerden (1933): Die Wellengleichung des Elektrons in der allgemeinen Relativitätstheorie

Leopold Infeld and Bartel van der Waerden (1933). Die Wellengleichung des Elektrons in der allgemeinen Relativitätstheorie. Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 380–401 (errata on p. 474).

15. Albert Einstein, Leopold Infeld and Banesh Hoffmann (1938): The Gravitational Equations and the Problem of Motion

Albert Einstein, Leopold Infeld and Banesh Hoffmann (1938). The Gravitational Equations and the Problem of Motion. The Annals of Mathematics, 39: 65–100.

16. Erwin Schrödinger (1940): The General Theory of Relativity and Wave Mechanics

Erwin Schrödinger (1940). The General Theory of Relativity and Wave Mechanics. Wis-en natuurkundig Tijdschrift, 10: 2–9.

The Quantization of Gravity

17. Without New Difficulties: Quantum Gravity and the Crisis of the Quantum Field Theory Program

Alexander Blum

In the years 1926–1928, immediately following the creation of matrix and wave mechanics, the protagonists of this development elaborated and expanded the techniques of the new quantum mechanics, so as to apply them to field theories.[1] This work culminated in the theory of interacting quantum electrodynamics (QED),[2] published in 1929 by Werner Heisenberg and Wolfgang Pauli (Heisenberg and Pauli 1929a). This paper, which deals mainly with the canonical quantization of both the electromagnetic and the matter-wave fields, famously contains a brief nod to gravitational theory, which is nowadays often quoted jokingly due to its seemingly naive optimism:

We further note that a quantization of the gravitational field, which appears to be necessary for physical reasons, should be also possible using a formalism entirely equivalent to the one used here without new difficulties. (Heisenberg and Pauli 1929a, 3)

Now this quote doesn’t sound half as optimistic if the emphasis is put on new. For the theory of quantum electrodynamics which Heisenberg and Pauli had just constructed was replete with difficulties. Three of these difficulties will play an essential role in our story:

The theory led to divergent expressions for the energies of stationary states. Even worse, J. Robert Oppenheimer, who was working with Pauli in Zurich at the time, could also show that the differences between these energies (i.e., the actually observed frequencies of spectral lines) came out infinite (Oppenheimer 1930).

In order to write down a Lorentz-invariant Lagrangian for the interacting electromagnetic and matter-wave fields, it was necessary to work with the electromagnetic potentials \phi and not just with the fields. But the Lagrangian does not contain then a time derivative of the electric potential \phi _0, so that there is no corresponding canonical momentum variable, preventing the straightforward implementation of canonical commutation relations.

The theory was not manifestly covariant due to the use of equal-time commutation relations. These allowed for a close analogy with the canonical commutation relations of non-relativistic quantum mechanics, but, by singling out time, destroyed manifest covariance. The Lorentz invariance of the theory thus had to be (and was) proven in a rather roundabout manner.

All three difficulties also played an important role in early work on the quantization of the gravitational field. I will be referring to these three difficulties as the divergence, the momentum, and the quantization difficulty, respectively. I will begin by discussing the momentum difficulty.

This difficulty was initially solved by Heisenberg and Pauli by adding to the Lagrangian additional terms, which contained a time derivative of \phi _0 and were proportional to a parameter \epsilon, which was supposed to be set to zero in the final expressions for physical quantities. This procedure was viewed as rather artificial from the start. Heisenberg, who had cooked up the method,[3] described it as a “very crude trick.”[4] Heisenberg consequently devised a new method for Heisenberg and Pauli’s second paper on QED (Heisenberg and Pauli 1929b).[5] This method relied on the notion of the gauge invariance of the theory of coupled electromagnetic potentials and Dirac matter waves \psi, which Weyl had only recently introduced (see the preceding part), that is, the invariance under a substitution

(17.1)

(17.2)

where \chi (x) is an arbitrary space-time function. Heisenberg’s idea was the following: The field \phi _0 was simply not quantized, thereby eliminating the need for a canonically conjugate momentum variable in order to construct the canonical commutation relation. \phi _0 was then simply a (c-number) function of space-time and could be set to zero, due to the well-known underdetermination of the electromagnetic potential. This brought with it a new difficulty, however: With \phi _0 set to zero, the equation of motion for \phi _0 (which is simply the first Maxwell equation, or Coulomb’s law, \mathrm {div} \vec{E} = \rho) no longer resulted from the variation of the Lagrangian and the dynamical problem was underdetermined. The equation of motion could also not simply be added as an operator identity, because it would imply non-vanishing commutation relations between matter and electromagnetic field operators, in contradiction with the canonical commutation relations.

Heisenberg now realized that one had not exploited the full gauge invariance by setting \phi _0 = 0. There was still a residual gauge symmetry, since if the function \chi doesn’t depend on time x_0, the transformation 17.2 leaves \phi _0 unaltered. To this residual symmetry now corresponded an operator that commutes with the Hamiltonian, that is, a conserved physical quantity. The conserved quantity corresponding to the residual gauge symmetry turned out to be

(17.3)

One could thus first solve the dynamical problem without the first Maxwell equation and then pick those solutions for which C=0, that is, for which the first Maxwell equation was fulfilled at some initial time. There was thus a Nebenbedingung (subsidiary condition) on the initial quantum state \varphi, which had to fulfill the equation C \varphi = 0. Since C commutes with the Hamiltonian, this condition on the initial state would propagate, and the first Maxwell equation would always be fulfilled, without actually being an operator identity.

The central difficulty with this new method was the apparent lack of Lorentz invariance: There was now no commutation relation for the zero component of the electromagnetic potential four-vector, and hence the commutation relations were no longer covariant — this was in addition to the difficulty of the equal-time commutators, that is, the quantization difficulty. Heisenberg and Pauli convinced themselves that “all statements about gauge invariant quantities [...] fulfill the demand of relativistic invariance,” but the proof they presented was highly problematic.[6]

This was the state of affairs for the momentum difficulty, when Pauli finally turned to the quantization of gravity. The immediate stimulus appears to have been the reading of Weyl’s paper on the interacting theory of Dirac matter waves, electromagnetic potentials and the gravitational field (Chapter 12).[7] He dismissed Weyl’s attempt to solve the problem of the negative energy states by using massless Weyl two-spinors instead of Dirac spinors, especially because he did not share Weyl’s hope that gravitational effects might be able to generate the mass term:

The hope of finding a replacement for the mass term in gravitational theory appears illusory to me; the gravitational effects will always be much too small numerically. (Hermann, Meyenn, and Weisskopf 1979, 519)

But Pauli realized that Weyl’s general scheme of coupling spinors to a curved space-time metric, by expressing the latter in terms of tetrads, would also work for regular, massive Dirac four-spinors.[8] Reformulated in this manner, Weyl’s field theory provided a natural extension of the field theory of interacting matter and electromagnetic waves that Heisenberg and Pauli had quantized for their QED. It thereby provided the obvious starting point for extending and completing their work by also including the gravitational field. Pauli concluded his remarks on Weyl’s paper by stating:

What now interests me most, is the question of how to quantize the \mathbf {e}(\alpha ) [the tetrads] themselves in your gravitational theory. (Hermann, Meyenn, and Weisskopf 1979, 520)

This was the question which he set his assistant Léon Rosenfeld. Rosenfeld soon discovered that one encountered problems akin to the momentum difficulty of QED.[9] For example, the canonical momenta for the time-components of the four tetrad basis vectors identically vanished. In further pursuing this question, Rosenfeld obtained several interesting results, which he published in a lengthy paper in 1930 (Chapter 18).[10] First of all, he was able to demonstrate in general that the momentum-type difficulties were the result of the invariance of the Lagrangian with regard to certain groups, the gauge group for the case of electrodynamics, the group of general coordinate transformations for general relativity, and the local Lorentz symmetry of the tetrad formulation. He then went on to devise a general method for dealing with such difficulties, a method which also managed to bypass the difficulties of Lorentz covariance encountered in the (second) Heisenberg-Pauli scheme.

In order to sketch Rosenfeld’s method, I will focus on the simple case of QED, which is the only example he really worked out to the end. The general idea was to also introduce canonical commutation relations for, and thereby to quantize, the electric potential \phi _0: Rosenfeld simply assumed that there existed a momentum operator \mathfrak {P}^4 that did in fact obey the canonical commutation relation with the electric potential. In order to do this two points needed to be addressed.

First, this meant that the Hamiltonian HH would contain a term \dot{\phi }_0 \mathfrak {P}^4. In order to have the Hamiltonian expressed solely in terms of the canonical variables, one would have to express \dot{\phi }_0 in terms of the canonical momenta (and the canonical coordinates, that is, the components of the potential). But the original Lagrangian did not depend on \dot{\phi }_0, and consequently \dot{\phi }_0 did not show up in the expressions relating the time derivatives of the field and the canonical momenta. This implied that \dot{\phi }_0 could be an arbitrary function of space and time without contradicting the defining equations for the canonical momenta. Rosenfeld thus set \dot{\phi }_0 in the Hamiltonian equal to an arbitrary function \lambda (x), which then consequently showed up in the equations of motion for the four-potential. In particular the equation of motion for the electric potential was simply of the form \dot{\phi }_0 = \lambda, ensuring the self-consistency of the approach. A specification of \lambda then corresponded to choosing a gauge.

The second point was that one still needed to take account of the fact that the momentum conjugate to the electric potential was actually zero. Rosenfeld introduced a Heisenberg-Pauli type Nebenbedingung on the state, demanding that \mathfrak {P}^4 \varphi = 0. And in order to have this condition propagate in time, an additional condition needed to be imposed

(17.4)

One thus obtained the same Nebenbedingung that Heisenberg and Pauli had imposed, ensuring the validity of Coulomb’s law. No further conditions were necessary, since C was a constant, as Heisenberg and Pauli had already shown.

Since Rosenfeld’s scheme was essentially equivalent to the Heisenberg-Pauli method for the specific choice \lambda = 0 (although it should be noted that it was not actually necessary to specify \lambda at all), Rosenfeld’s work in the context of QED could simply be viewed as a proof of the covariance of that method, and this is how he later presented it,[11] resorting to the simpler Heisenberg-Pauli method for actual calculations (Rosenfeld 1932).

In any case, it was a whole different approach that came to be the standard method in the QED of the 1930s and 1940s, due to Enrico Fermi, which was based on taking the (Lorenz) gauge condition (as opposed to Coulomb’s law) and its time derivative as conditions on the wave function (Fermi 1932).[12] It involved the use of a modified Lagrangian that was not gauge invariant and only returned the Maxwell equations if the Lorenz gauge was imposed. Rosenfeld argued against the Fermi approach in his paper, on account of its lacking gauge covariance, demonstrating that his own method was in fact gauge covariant. But this was not a very strong argument at the time, and also Rosenfeld could field no arguments as to why one should attach much weight to gauge covariance in the first place.

How then did Rosenfeld’s method fare in the context of gravitation? It did indeed prove applicable, in particular because it was generalizable to the case where the constraints on the canonical momenta were more complicated than the mere vanishing of some of the components, as was the case for the constraints arising from the local Lorentz covariance of the tetrad formalism. But he stopped short of actually constructing a canonical Hamiltonian for the Weyl-Fock field theory. For this, he would have had to solve the equations defining the canonical momenta for the time derivatives of the fields (i.e., of the tetrads), in order to end up with a Hamiltonian that only contained canonical coordinates and momenta, as well as the arbitrary functions of the type \lambda.

Why did Rosenfeld not do this? This question remains unanswered. Years later, Pauli remarked that Rosenfeld’s work was “not satisfactory in all aspects, because he had to introduce certain additional conditions, which no one could really understand.”[13] It is entirely unclear, however, what Pauli was referring to. Rosenfeld’s work remained for the next two decades an anomaly, until the late 1940s, when the systematic study of constrained Hamiltonian dynamics picked up steam, and several of Rosenfeld’s results were re-discovered, as will be discussed in the final part of this book. One additional, simple result of Rosenfeld’s, unrelated to the momentum difficulty, has stood the test of time: The necessity of quantizing the gravitational field with commutators, that is, the realization that gravitons must be bosons. Ten years before Pauli’s formulation of the spin-statistics theorem, this was a non-trivial result.

One thing is certain: Solving the momentum relations for the time derivatives of the fields is a complicated business. This would not be the last instance of physicists convincing themselves that methods from QED also worked for gravity in principle, without filling in the details. In any case, it turned out to be unnecessary for Rosenfeld’s second work on the quantum theory of gravity, which he completed in the same year (Chapter 19). This second paper, much more than the first, which certainly started out as an attempt to quantize gravity, is an attempt to better understand the difficulties of QED by studying whether similar difficulties appeared in other quantum field theories. The difficulty in question here was the first difficulty mentioned above, the divergence difficulty, to which we will now turn.

It was initially unclear whether the divergence of the self-energy of the electron in QED was simply an inheritance from the classical theory, where it was well-known that the notion of a point electron was highly problematic, due to the infinite electromagnetic mass associated with such an object. Supposedly Heisenberg had suggested[14] that one should study the case of the gravitational self-energy of the photon, since there were no singularities in the corresponding classical theory of an electromagnetic wave interacting with the gravitational field. In any case, Pauli egged Rosenfeld on to pursue this problem.[15]

The electromagnetic self-energy of the electron had been calculated perturbatively, expanding in terms of the coupling constant, the electron charge e. The divergent expression then arose at second order in perturbation theory. Consequently, Rosenfeld performed his calculation of the gravitational self-energy of the photon in the approximation of a weak gravitational field. Such an approximation had, of course, been worked out by Einstein immediately after the formulation of general relativity (see the first part of this book). Einstein had started from the assumptions that the deviations \gamma _{\mu \nu } from the flat space-time metric were numerically small (compared to the elements of the flat space-time metric itself, which are of order 1). Rosenfeld now factored out a co-efficient \epsilon, the square root of Einstein’s gravitational constant, the deviations from the flat space-time metric thus now being of the form \epsilon \gamma _{\mu \nu }, Rosenfeld’s \gamma now being a dimensionful quantity. This explicitly made Einstein’s approximation equivalent to an expansion in terms of the gravitational coupling constant.

As Einstein had further remarked, solving the field equations of general relativity in this approximation was basically equivalent to calculating retarded potentials in electrodynamics. This also meant that all the techniques developed for QED could be taken over to the quantization of (linearized) gravity in a very straightforward manner. In particular, there was no need for Rosenfeld’s new method of dealing with the momentum difficulty.[16] The Fermi method could be taken over from QED and applied directly to gravity, the Lorenz gauge condition being replaced by the analogous coordinate condition imposed by Einstein

(17.5)

where \gamma ^\prime _{\mu \nu } is the trace-reverse of \gamma _{\mu \nu }. Following Fermi, Rosenfeld constructed a Lagrangian which was not simply the linearized Einstein-Hilbert Lagrangian, and hence not even approximately invariant under general coordinate transformations. The most straightforward way to arrive at the Rosenfeld Lagrangian is to linearize the first-order Einstein-Hilbert Lagrangian, impose the above coordinate condition and then drop a total divergence. Rosenfeld presented no derivation of his Lagrangian, the only important point being that it returned the correct linearized field equations.

In this framework, Rosenfeld then calculated (to second order in \epsilon) the gravitational field energy in the absence of free gravitons, where the operator \gamma _{\mu \nu } can be expressed solely in terms of the electromagnetic field operators. He obtained two divergent terms, one constant and one proportional to the number of photons. The second term was interpreted as the gravitational field energy of a photon, and its divergent nature was subsequently cited as proof for the quantum nature of the self-energy divergence, independent of the presence of singularities in the corresponding classical theory.

There remained the suspicion that this divergent energy was merely the coupling of the gravitational field to the infinite zero-point energy of the electromagnetic field. This question was investigated soon after by Jacques Solomon, who redid Rosenfeld’s calculation after first eliminating the zero-point energy in the Hamiltonian of the electromagnetic field by imposing normal ordering on the annihilation and creation operators (Solomon 1931). Solomon showed that while the constant divergent term was eliminated in this manner, the photon self-energy remained divergent. But he had only imposed normal ordering on the level of the electromagnetic Hamiltonian and this normal ordering did not survive when calculating the energy of the gravitational field. I therefore think that it can still be argued that Rosenfeld’s divergence was merely the result of the coupling to the zero-point energy.

In any case, when Rosenfeld’s calculation was redone again by Bryce deWitt (né Seligman) in his unpublished PhD thesis (DeWitt 1949), now using modern covariant techniques, he obtained a vanishing photon self-energy, as should be expected from considerations of gauge invariance alone. DeWitt gave no real argument as to where Rosenfeld had gone wrong, which is quite understandable considering that almost all the calculations from the 1930s were considered obsolete when renormalized quantum field theory was developed in the late 1940s.

The question of the photon self-energy aside, Rosenfeld was still interested in the general question of the quantization of gravity and supplemented his paper with a section on the quantization of the linearized gravitational field also for the case of free gravitons. His treatment was, however, quite sketchy. Bypassing the canonical commutation relations, Rosenfeld went straight to an explicit expression for the operator \gamma _{\mu \nu } in terms of graviton annihilation and creation operators, by expanding in terms of transverse gravitational wave solutions of the free field equation. These transverse waves had been identified by Einstein as the only physical gravitational waves, carrying energy and propagating with the speed of light. Rosenfeld concluded with a brief look at graviton-photon scattering in this linear quantum theory of gravity.

The quantization of linearized GR wasn’t worked out in detail until 6 years later, in the PhD work of Soviet physicist Matvei Bronstein (Chapter 20).[17] It is unclear whether he was aware of Rosenfeld’s work,[18] but he did not really need to build on it. Bronstein had an independent program of formulating linearized GR as close to electrodynamics as possible, with \gamma _{\mu \nu } treated as a regular field in flat Minkowski space-time (as opposed to as a deviation from the flat space-time metric). He could then straightforwardly apply the canonical, equal-time field quantization procedure developed by Heisenberg and Pauli for QED (along with the Fermi method for dealing with the coordinate condition, which Bronstein now explicitly identified as a gauge condition) to all components of the linearized gravitational field, not only to those that corresponded to freely propagating waves.

These additional components became important when treating the interaction with matter. Here, Bronstein built on Fock’s work, expressing Fock’s generally relativistic Dirac equation, and in particular the tetrads, in terms of the \gamma _{\mu \nu }. Bronstein could thereby not only calculate the emission and absorption coefficients for transverse gravitons, but also show how the non-transverse gravitons led to a Newtonian gravitational interaction between electrons, in the same way that the longitudinal (and time-like) photons of QED led to the Coulomb interaction.

But Bronstein’s work is less known for his detailed treatment of the quantum theory of linearized gravity, and more for his musings on the full quantum theory of gravity. Bronstein transferred Bohr and Rosenfeld’s analysis of the measurability of field quantities in QED (Bohr and Rosenfeld 1933) to quantum gravity. Bohr and Rosenfeld had concluded that there were no further restrictions on the measurability of the electromagnetic field beside those arising from the canonical uncertainty relations, just as in quantum mechanics. This conclusion had centrally rested on the assumption that the test body used to measure the field was macroscopic (i.e., its atomistic nature could be disregarded) and could be given an arbitrarily high charge density. Bronstein argued that, although the close relation between QED and the quantum theory of gravity allowed adapting almost the entire Bohr-Rosenfeld analysis, the final conclusion would be different, because general relativity did not allow for arbitrarily dense bodies. Rather, there was a fundamental limit from the fact that a body could not become smaller than its gravitational (Schwarzschild) radius. There were thus absolute limits on the measurability of the gravitational field and a quantum theory of gravity would require major conceptual changes.

Now Bronstein’s reasoning clearly shows great physical intuition. He was the first to realize the essential difficulties inherent in constructing a quantum theory of gravity and that such a theory was not to be had without fundamental conceptual innovations. His argument is fielded to this day as an elementary demonstration of the problem that is quantum gravity. Why then was Bronstein’s argument hardly received by his contemporaries? To understand why it didn’t make much of a splash at the time, one needs to look at its weaknesses in somewhat more detail.

One objection was already raised at Bronstein’s thesis defense by Wladimir Fock:[19] The absence of a fundamental restriction on the charge density was hardly a desirable feature of quantum electrodynamics. Rather, it allowed for a theory such as QED with point charges that had an apparently infinite charge density, resulting in the divergence difficulties discussed above. Physicists at the time were thus much more inclined to believe that revisions in the concepts of space and time would come from a reformed (quantum) electrodynamics, where a quantity of the order of the classical electron radius would play an essential role. The radical changes that Bronstein was envisioning were thus expected to come in at much larger length scales. It was only when the difficulties of QED (and then also of nuclear physics) had been solved — without radical changes in the underlying space-time theory—that Bronstein’s arguments could be viewed as convincing.[20]

But there is a more intrinsic difficulty in Bronstein’s argument as well. The aim of the Bohr-Rosenfeld analysis had been to take the established mathematical framework of QED and prove that it was internally consistent, that is, that there were no fundamental limitations on the observability of the electromagnetic field components, which would have been in contradiction with the initial choice of these components as the dynamical (quantum) variables. The Bohr-Rosenfeld analysis was thus a procedure to be applied to an already formulated quantum theory, in order to see whether it contained inherent contradictions. Consequently, in the original Russian article based on his thesis (Bronstein 1936), Bronstein applied this analysis to his linearized quantum gravity (which was a fully formulated quantum field theory), and not to an as yet hypothetical full quantum theory of gravity. He then voiced the hope that his argument would also carry over to the full theory, even though his analysis could not be applied to it, since it had not yet been formulated.

Somewhere between writing up his thesis and publishing the German-language article reprinted here, Bronstein must have realized that this argument was fundamentally flawed. There was nothing in the classical, linearized theory that prevented a body from being smaller than its gravitational radius. And so the linearized quantum theory was just as consistent as QED. In the German article, he thus presented the Bohr-Rosenfeld argument as an argument against the coherence of a quantum field theory based on the full non-linear theory of gravity, disregarding the obvious logical difficulty of showing the inconsistency of a theory that didn’t even exist yet.

This criticism of Bronstein’s argument was raised by Jacques Solomon (Chapter 21), but not without bringing forth a further argument against the possibility of constructing a quantum field theory based on the full non-linear theory.[21] Solomon’s argument was based on a recent proof by Nathan Rosen on the non-existence of non-singular plane wave solutions in full GR (Rosen 1937).[22] Now, all quantum field theories of the time were based on an expansion of the field quantities in plane waves. Even if Heisenberg and Pauli’s starting point had been commutation relations for the field quantities themselves, as soon as they went to the level of representing these field quantities as operators on a Hilbert space, they had, as they grudgingly admitted, no choice but to expand them in plane wave solutions of the free field equations and treat the expansion coefficients as annihilation and creation operators on occupation number space. From the absence of plane wave solutions, Solomon concluded that the present formalism of field quantization was not compatible with the non-linear theory of gravitation, a weaker claim than Bronstein’s, but one that found at least one interested listener in Pauli, as we shall see later.

But of course both Bronstein’s and Solomon’s thoughts on the difficulties of the non-linear quantum theory of gravity could be regarded as nitpicking, as long as the difficulties of QED (and thereby also of the quantum theory of linearized gravity) remained unsolved. The attempts to solve these difficulties, in as far as they were difficulties of quantum field theory in general, of course also impacted the pursuit of quantum gravity. It is well-known that the most fundamental difficulty of QED, the divergence difficulty, was not solved until the late 1940s. But progress was being made already in the 1930s on lesser difficulties, in particular on the lack of manifest covariance due to the use of equal time commutators, what I have called the quantization difficulty. In the last part of this essay, I will focus on this third and final difficulty.

While certainly not the most pressing difficulty at the time, there were enough physicists who believed that formulating quantum theory in a more overtly relativistic manner was a worthwhile endeavor. Two important relativistic quantization procedures were devised in order to replace equal-time commutators in the first half of the twentieth century.[23] In both cases, Paul Dirac played an essential role. The union of relativity and quantum theory was a leitmotif in his work from the very start, when he attempted to make Heisenberg’s matrix mechanics relativistic by turning time into a non-commuting matrix (Dirac 1926). This was followed by the Dirac equation in 1928 (Dirac 1928) and then by several hugely influential papers in 1932/33, in which he laid the foundations for several new, relativistically invariant quantization techniques. We begin by discussing his 1933 paper on “The Lagrangian in Quantum Mechanics” (Dirac 1933).

Dirac’s general idea was to formulate quantum mechanics not in terms of states (wave functions) propagating in time, but in terms of transition amplitudes from the state at one time to the state at a later time. Dirac called these transition amplitudes “transformation functions,” since he conceptualized them as generating canonical transformations from the canonical coordinates at one time to those at a later time. In the final section, he also hinted at how to generalize this idea to quantum field theory, where the initial and final times would be replaced by an arbitrary (not necessarily space-like) three-dimensional hypersurface of space-time, which formed the integration boundary of the classical action. He dubbed the field theoretic amplitudes relating the canonical (field) coordinate values on different points on this hypersurface[24]“generalized transformation functions.” Dirac’s ideas, however, remained very vague and were not worked out in any detail by him.

Dirac’s idea was taken up by Paul Weiss,[25] who realized that the procedure of replacing initial and final times by general three-dimensional boundary hypersurfaces might also be applicable to the canonical quantization procedure (Weiss 1936). At first sight, the canonical quantization procedure seemed to rest essentially on the singling out of time, since the momentum canonically conjugate to a field variable \phi appearing in the Lagrangian L was defined as \frac{\partial \mathcal {L}}{\partial \left(\partial _t \phi \right)}. This was one of the reasons why Dirac had based his sketch on the Lagrangian formalism, where the canonical momenta do not enter. Weiss now noted that the canonical momentum could also be defined in another way. If one reads the definition of the canonical momentum as the time component of the four-vector \frac{\partial \mathcal {L}}{\partial \left(\partial _{\mu } \phi \right)}, then this can be generalized by taking the canonical momentum as the component of the four-vector orthogonal to the hypersurface forming the integration boundary of the action, on which the values of the field variables are fixed.

Weiss initially, like Dirac, took the boundary hypersurface to be closed. So, while one could locally determine the direction w perpendicular to the surface and describe points on the surface by three parameters u_i, this coordinate system (dubbed “natural” by Weiss) could not be regular throughout space-time. Still, one could write down commutators for the field operators, which were exactly of the form of the Heisenberg-Pauli equal-time commutators, except that the arguments of the field operators were now not the three spatial coordinates, but rather the three surface parameters u_i. Further, a Hamiltonian could be constructed in the usual fashion, using the generalized canonical momenta, and be shown to generate the “time evolution” (i.e., the evolution for increasing w) of the field operators.

In his second, and most influential paper (Chapter 22) , Weiss could show that the generalization of the commutation relations need not necessarily occur only at the quantum level, where it smacked of an ad-hoc modification of the well-established canonical quantization procedure. Instead, he could demonstrate that his generalized commutation relations corresponded already at the classical level to generalized Poisson brackets, and that his method was thus a straightforward generalization of the usual canonical quantization method.

This reconceptualization brought with it a slight modification of the original method. If the generalized Poisson brackets are supposed to be invariant under canonical transformations, in particular under time evolution, as are their point-mechanical counterparts, they can only sensibly be defined on some initial space-like hypersurface (on which then the values of both the field variables and the canonical momenta are given). Weiss thus dropped his original approach of having the commutation relations defined on an entire closed hypersurface, and replaced it with commutation relations on some (initial) space-like hypersurface, which corresponded to the classical, generalized Poisson bracket. This quantization procedure provided a covariant generalization of the equal-time commutators that, however, still stuck closely to the canonical scheme. We will discuss its further use in the last part of this book, which discusses the beginnings of what would come to be known as the canonical quantization approach to quantum gravity.

The other quantization procedure that went beyond the Heisenberg-Pauli equal-time commutators was not a direct generalization of the canonical quantization procedure, as Weiss’s method was. At its core were not the commutation relations between canonically conjugate field variables, but rather the commutation relations between two field variables at arbitrary different points in space-time. Such covariant commutators had first been used in 1928 by Jordan and Pauli for their quantization of the free electromagnetic field (Jordan and Pauli 1928). They were constructed by expanding the electromagnetic field variables in terms not of time-independent spatial modes, but in terms of time-dependent, propagating plane waves, which were solutions of the (free) equations of motion. Imposing the usual annihilation-creation operator commutation relations on the expansion coefficients, and then re-summing, the covariant commutation relations were obtained.

Since the construction was based on an expansion in terms of solutions of the wave equation, the commutation relations were automatically compatible with the equations of motion, a necessary requirement for any equation relating the field at two time-like (or, for the case of electrodynamics more importantly, light-like) separated points. Indeed, the commutators themselves actually had to be (singular) solutions of the equations of motion. This immediately became intractable once one was dealing with interacting fields and non-linear equations of motion, which is why one year later, Pauli reverted to the use of equal-time commutators.

Covariant commutation relations were brought back into the game by Dirac in 1932. In an attempt to relaunch QED, he had effectively reconstructed Heisenberg-Pauli quantum field theory, only now in the interaction representation (Dirac 1932). Dirac was much ridiculed by Pauli for having merely produced an equivalent theory, despite his grandiloquent claims to the contrary. In a letter to Dirac of 11 September 1932,[26] Pauli wrote:

Your remarks on quantum electrodynamics [...] were—to put it mildly—certainly no masterpiece. After a confused introduction, consisting of sentences that were only halfway understandable because they were only halfway understood, you finally arrive at results for a simplified, one-dimensional example that are identical with those obtained by applying the formalism of Heisenberg and myself to this example. (The identity is immediately recognizable and was then calculated in too complicated a manner by Rosenfeld.) This conclusion of your work stands in contradiction to the more or less clearly voiced claims in the introduction that you would somehow be able to make a better quantum electrodynamics than Heisenberg and myself. (Hermann, Meyenn, and Weisskopf 1985, 115)

But when Dirac, together with Vladimir Fock and Boris Podolsky, actually applied his interaction representation formulation of quantum field theory to the full electrodynamics (Dirac, Fock, and Podolsky 1932), and not just to the toy example of the first paper, Pauli realized that it had a huge advantage: All the non-trivial dynamics due to the interaction between charged matter and electromagnetic field were relegated to the time evolution of the state vector. In other words: Also the second inhomogeneous Maxwell equation, the Ampère-Maxwell law, was satisfied only through the action of the field operators on the wave function and not as an operator identity. This implied that the electromagnetic field operators still obeyed the free field equations and consequently the covariant commutation relations of Jordan and Pauli. On 2 June 1933, Pauli wrote to Heisenberg:

As time goes by, I find myself liking the work of Dirac, Fock and Podolsky more and more. It is funny that there the commutation relations of vacuum electrodynamics do not change when particles are present, even for t \ne t^\prime. (Hermann, Meyenn, and Weisskopf 1985, 167)

But the method of quantizing by imposing covariant commutation relations was still an isolated technique, applied only to the Maxwell field, far-removed from the generality of the canonical quantization approach, which could be applied to any classical field theory given in Lagrangian or Hamiltonian form. It was extended in 1938 by Stueckelberg to encompass massive scalar fields, again by explicitly expanding the field operators in terms of solutions of the wave equation (in this case the Klein Gordon equation) (Stückelberg 1938a), and then also to massive vector (Proca) fields (Stückelberg 1938b). In 1939, finally, Markus Fierz gave a general expression for the covariant commutation relations for fields with arbitrary spin (Fierz 1939).[27] The covariant quantization method thereby became a full-fledged quantization method in its own right, applicable to any Lorentz-covariant field theory.

What did this mean for the quantization of gravity? General Relativity was, of course, not a Lorentz-invariant field theory, but as we have seen, the linearized theory could be thought of in this manner. And indeed, Fierz together with Pauli could show that the linearized gravitational field fit into Fierz’s scheme, namely as a massless spin 2 field (Chapter 23).

This was a major conceptual change: Bronstein (and Rosenfeld) had started from the full theory of general relativity and obtained a Lorentz-covariant field theory as an approximation. Pauli and Fierz now showed that one could construct the field theory of linearized gravity equally well without any reference to general relativity: it was the unique Lorentz-covariant field theory of a massless spin 2 field. Also the transformation properties of the metric perturbations \gamma _{\mu \nu } arose, without any reference to general coordinate transformations, as the gauge transformation properties of a massless field, a straightforward generalization of electrodynamics.

Initially, of course, this change was hardly consequential: The quantum field theory of linearized gravity had already been worked out, and neither the use of covariant quantization methods, nor the reconceptualization in terms of a spin 2 field made any difference to, say, Bronstein’s formulation. But it would eventually lead to a new way of thinking about the quantum theory of non-linear gravity. Pauli, when recapitulating his work with Fierz in his manuscript for the 1939 Solvay conference (which never took place due to the outbreak of World War II, but is reprinted in Meyenn (1993)), was still quite pessimistic concerning the prospects for the non-linear theory. In particular he cited Solomon’s argument, but with an interesting twist:

It is certainly a limitation of the quantum theoretical side of these considerations that one here contents oneself with the approximation in which the generally relativistic field equations are linear. This limitation is intimately connected with the well-known divergence difficulties of field theory. (Meyenn 1993, 901)

No further explanation of this connection is given, but I would suggest the following reading: In quantum electrodynamics it is the non-linear terms, that is, the terms coupling the electromagnetic field to the charged current of the electrons, that lead to the divergence difficulties. Pauli’s connection of the non-linearities of GR with the divergence difficulties of QED thus offered an entirely new way to look at these non-linearities: For Solomon, these were terms arising in a higher approximation to the full theory, spoiling the usual quantum field theoretical approach by eliminating the possibility of plane wave solutions. In contrast, the non-linear terms could now be considered as interaction terms, added on to the Lagrangian for the free spin 2 field, just like the interaction terms of QED were added on to the free field Lagrangians, with the sole difference that one was now talking about self-interactions of a single field.

As long as the divergence difficulties of QED remained unresolved, this hardly made a difference. But when covariant renormalization was developed in the late 1940s, it seemed plausible that the same techniques might also be used for the non-linear theory. By using the interaction representation, one could quantize the gravitational field in the same manner as the electromagnetic field, by separating the free spin 2 quantum (which had plane wave solutions, thereby circumventing Solomon’s objection) from its self-interaction. The self-interaction (as well as the interactions with the matter fields) could then be treated in the same manner as the interactions of QED, possible divergences being absorbed into the properties of the free field. Thus the program of covariant quantization was born, which, however, took off only somewhat after the period covered in this book. We thus see how the two major programs for the quantization of gravity, canonical and covariant quantization,[28] had their origins in the 1930s, in two attempts to get over the quantization difficulty of QED with, at this early stage, only weak and incidental connections to the specific problem of quantum gravity.

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——— (1928). “The Quantum Theory of the Electron.” Proceedings of the Royal Society of London: Series A, Containing Papers of a Mathematical and Physical Character. JSTOR, 610–24.

——— (1932). “Relativistic Quantum Mechanics.” Proceedings of the Royal Society, Series A 136 (829): 453–64.

——— (1933). “The Lagrangian in Quantum Mechanics.” Physikalische Zeitschrift Der Sowjetunion 3 (1): 64–72.

Dirac, Paul A. M., Vladimir A. Fock, and Boris Podolsky (1932). “On Quantum Electrodynamics.” Physikalische Zeitschrift Der Sowjetunion 2 (6): 468–79.

Fermi, Enrico (1929). “Sopra L’elettrodinamica Quantistica.” Rendiconti Lincei 9: 881–87.

——— (1932). “Quantum Theory of Radiation.” Review of Modern Physics 4: 87–132.

Fierz, Markus (1939). “Über Die Relativistische Theorie Kräftefreier Teilchen Mit Beliebigem Spin.” Helvetica Physica Acta 12: 3–37.

Gorelik, Gennady E., and Victor Ya. Frenkel (1994). Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties. Basel: Birkhäuser.

Heisenberg, Werner, and Wolfgang Pauli (1929a). “Zur Quantendynamik Der Wellenfelder.” Zeitschrift Für Physik 56: 1–61.

——— (1929b). “Zur Quantentheorie Der Wellenfelder. Band Ii.” Zeitschrift Für Physik 59 (3–4): 168–90.

Hermann, Armin, Karl von Meyenn, and Victor F. Weisskopf, eds. (1979). Wolfgang Pauli: Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.a. Vols. Vol. 1: 1919–1929. New York: Springer.

———, eds. (1985). Wolfgang Pauli: Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.a. Vols. Vol. 2: 1930–1939. New York: Springer.

Jordan, Pascual, and Wolfgang Pauli (1928). “Zur Quantenelektrodynamik Ladungsfreier Felder.” Zeitschrift Für Physik 47: 151–73.

Kobzarev, Igorʹ Ju., ed. (1985). Einsteinovski Sbornik, 1980–1981. Moscow: Nauka.

Meyenn, Karl von, ed. (1993). Wolfgang Pauli: Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.a. Vols. Vol. 3: 1940–1949. Berlin: Springer.

———, ed. (2001). Wolfgang Pauli: Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.a. Vols. Vol. 4, Part 3: 1955–1956. Berlin: Springer.

Misner, Charles W. (1957). “Feynman Quantization of General Relativity.” Reviews of Modern Physics 29 (3): 497–509.

Oppenheimer, J. Robert (1930). “Note on the Theory of the Interaction of Field and Matter.” Physical Review 35 (5): 461–77.

Rickles, Dean, and Alexander Blum (2015). “Paul Weiss and the Genesis of Canonical Quantization.” European Physical Journal H 40 (4–5): 469–87.

Rosen, Nathan (1937). “Plane Polarised Waves in the General Theory of Relativity.” Physikalische Zeitschrift Der Sowjetunion 12: 366–72.

Rosenfeld, Leon (1932). “La Théorie Quantique Des Champs.” Annales de L’Institut Henri Poincaré 2 (1): 25–91.

Rovelli, Carlo (2004). Quantum Gravity. Cambridge: Cambridge University Press.

Rueger, Alexander (1992). “Attitudes Towards Infinities: Responses to Anomalies in Quantum Electrodynamics, 1927–1947.” Historical Studies in the Physical and Biological Sciences 22: 309–37.

Salisbury, Donald (2009). “Léon Rosenfeld and the Challenge of the Vanishing Momentum in Quantum Electrodynamics.” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 40 (4): 363–73.

Schweber, Silvan S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton: Princeton University Press.

Solomon, Jacques (1931). “Nullpunktsenergie Der Strahlung Und Quantentheorie Der Gravitation.” Zeitschrift Für Physik 71 (3–4): 162–70.

Stachel, John (1999). “The Early History of Quantum Gravity (1916–1940).” In Black Holes, Gravitational Radiation and the Universe, 525–34. Springer.

Stückelberg, Ernst C. G. (1938a). “Die Wechselwirkungskräfte in Der Elektrodynamik Und in Der Feldtheorie Der Kernkräfte: Teil 1.” Helvetica Physica Acta 11 (225–244).

——— (1938b). “Die Wechselwirkungskräfte in Der Elektrodynamik Und in Der Feldtheorie Der Kernkräfte: Teil 1fte in Der Elektrodynamik Und in Der Feldtheorie Der Kernkräfte: Teil 2 Und 3.” Helvetica Physica Acta 11: 299–328.

Weiss, Paul (1936). “On the Quantization of a Theory Arising from a Variational Principle for Multiple Integrals with Application to Born’s Electrodynamics.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 156 (887): 192–220.

Footnotes

[1] For the early history of quantum field theory, see, e.g., Cini (1982); Darrigol (1982; 1986) and the first two chapters of Schweber (1994). The difficulties of early quantum field theory are discussed in Rueger (1992). An overview that places the development of quantum field theory in the larger context of the development of quantum mechanics can be found in a chapter by Christoph Lehner and the author in a forthcoming volume on the genesis of quantum mechanics.

[2] As opposed to free quantum electrodynamics on the one hand (Jordan and Pauli 1928), and quantum radiation theory, i.e., a theory of transverse photons, on the other (Dirac 1927).

[3] See a letter from Pauli to Niels Bohr, 16 January 1929. All the Pauli letters from the 1920s are reprinted in Hermann, Meyenn, and Weisskopf (1979). All translations are by me.

[4] Heisenberg to Jordan, 22 January 1929, Archive for the History of Quantum Physics, MF 18, Section 002–024.

[5] See also a letter from Heisenberg to Pauli, 20 July 1929.

[6] Dirac in a letter to Rosenfeld from 6 May 1932 (Niels Bohr Archive, Copenhagen) (under)stated that he found it “difficult to understand,” and Rosenfeld concurred on 10 May (Dirac Papers, Churchill College, Cambridge) that Dirac was “right in not understanding” the “highly doubtful sentence” with which Heisenberg and Pauli concluded their proof.

[7] Letter from Pauli to Weyl, 26 August 1929.

[8] This is of course what Fock had already done. But Pauli was not aware of Fock’s work until somewhat later. See a letter to Ehrenfest from 29 September 1929.

[9] Letter from Pauli to Jordan, 30 November 1929.

[10] For a detailed discussion of this work, in particular with regard to its relation with later work on constrained Hamiltonian dynamics, see Salisbury (2009).

[11] Letter to Dirac, 21 May 1932, Dirac Papers, Churchill College, Cambridge.

[12] It should be noted that Fermi himself actually gave no indications as to how exactly the gauge condition should be interpreted in his original work (Fermi 1929). It was only Heisenberg and Pauli who interpreted it as a condition on the state, akin to their own Nebenbedingung, an interpretation which Fermi then adopted.

[13] ... nicht in jeder Hinsicht befriedigend war, da er gewisse zusätzliche Bedingungen einführen musste, die niemand richtig verstehen konnte. Letter to Oskar Klein, 25 January 1955, reprinted in Meyenn (2001).

[14] When and where is uncertain. Pauli in a letter to Rosenfeld (19 September 1930, but reprinted in Meyenn (1993)) speculates that it might have been during discussions in Copenhagen, but Heisenberg did not actually attend the 1929 spring conference in Copenhagen (8–15 April, see the timeline in Hermann, Meyenn, and Weisskopf (1979)), since he was on his world tour (from which he only returned in November, see the timeline in Cassidy (1992)).

[15] Interview of Léon Rosenfeld by Thomas Kuhn and John Heilbron, Niels Bohr Library and Archives, American Institute of Physics, College Park, MD, USA, https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4847-2 (accessed 21 July 2017).

[16] It of course also needs to be noted that there was no need to introduce the tetrads in this case, since no spinorial matter was involved.

[17] On Bronstein’s life and scientific work, see Gorelik and Frenkel (1994).

[18] He was certainly aware of Rosenfeld’s first paper: During Bronstein’s thesis defense (the protocol is reprinted in Kobzarev (1985, 317–320)), Igor Tamm recommended that he make reference to it, which he did in the published Russian paper. Rosenfeld’s second paper on linearized gravity is, however, never mentioned.

[19] The protocol of the defense has been published in Russian, see Footnote 18.

[20] Gorelik and Frenkel are somewhat dismissive of Fock’s objection, because they focus too narrowly on a specific attempt at introducing the electron radius into electrodynamics, the Born-Infeld theory, as did Bronstein in his reply to Fock. However, Fock explicitly stated that “the generalzation [of electrodynamics] to a non-linear theory is only beginning” (Kobzarev 1985, 318). He was thus referring to the general program and not merely to its specific, and problematic, implementation by Born and Infeld.

[21] In the secondary literature, e.g. in Stachel (1999), Solomon’s and Bronstein’s arguments have in general been lumped together.

[22] Rosen’s conditions were later seen as too strict, since he was also ruling out mere coordinate singularities (Bondi, Pirani, and Robinson 1959).

[23] Not counting Feynman’s path integral, which was not applied to the case of gravitation until the mid-1950s, see in particular Misner (1957).

[24] They should not be called transition amplitudes, since this would imply a notion of initial and final state. Maybe “correlation amplitudes” would be a good name.

[25] For a detailed discussion of Weiss’s life and work, see Rickles and Blum (2015).

[26] All Pauli letters from the 1930s are reprinted in Hermann, Meyenn, and Weisskopf (1985).

[27] The context of these works is the discovery of the putative Yukawa meson in 1937, see Blum (2014).

[28] See, e.g.,Rovelli (2004), Appendix B.

18. Léon Rosenfeld (1930): Zur Quantelung der Wellenfelder

Léon Rosenfeld (1930). Zur Quantelung der Wellenfelder. Annalen der Physik, 397: 113–152.

19. Léon Rosenfeld (1930): Über die Gravitationswirkungen des Lichts

Léon Rosenfeld (1930). Über die Gravitationswirkungen des Lichts. Zeitschrift für Physik, 65: 589–599.

20. Matvei Bronstein (1936): Quantentheorie schwacher Gravitationsfelder

Matvei Bronstein (1936). Quantentheorie Schwacher Gravitationsfelder. Physikalische Zeitschrift der Sowjetunion, 9: 140–157.

21. Jacques Solomon (1938): Gravitation et Quanta

Jacques Solomon (1938). Gravitation et Quanta. Journal de Physique et Le Radium, 9: 479–485.

22. Paul Weiss (1938): On the Hamilton-Jacobi Theory and Quantization of a Dynamical Continuum

Paul Weiss (1938). On the Hamilton-Jacobi Theory and Quantization of a Dynamical Continuum. Proceedings of the Royal Society of London, A169, 102–119.

23. Markus Fierz and Wolfgang Pauli (1939): On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field

Markus Fierz and Wolfgang Pauli (1939). On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field. Proceedings of the Royal Society of London, A173, 211–232.

Discreteness and Divergences

24. The Emergence of Quantum Geometry

Dean Rickles

A central theme of the previous selection of papers was the existence of a variety of divergences, especially the problems generated by the self-energy of elementary particles.[1] A common thread that emerges, persisting to this day, is that gravitation might be able to get some kind of foothold in the theory of elementary particles if it could be invoked to provide a physical (non-ad hoc) cutoff at wavelengths that do not conflict with current observations of experimental particle physics data.[2] Hence, Einstein’s old question about the role of gravity in elementary particle physics (mentioned in the introduction to part one of this volume) comes alive once again, though not in the way he envisaged it: gravity is rather “put to service” in elementary particle physics as a kind of external resource. However, the notion of a gravitationally induced cutoff took some time to form, and had to wait for a more thorough understanding of the non-linear aspects of the theory, which had remained largely buried under the more tractable linear approximation.[3] But this work can be seen as emerging from a well-worn path involving the usage of other cutoffs (minimum lengths) to tame the infinite behaviour of field theories—if gravity suggests a minimum length, then previous work on such ideas can be transferred across from context to context. The papers presented in this part, then, as with many others in this book, were not initially written with quantum gravity in mind; this was a connection only made later once the problem of quantum gravity itself had undergone several changes—not least the idea that quantum gravity involves quantum spacetime.

There were many early suggestions that gravity might act as some kind of divergence “regulator”. The divergences in question were those of QED, and meson theories, which were still, pre-WWII, a somewhat mathematically murky territory (quantum field theory still is, of course). The problem concerned the transitions between quantum states, during which time (a very short time, determined by the uncertainty relations) energy conservation is violated. The great hope for introducing gravitation into elementary particle physics was that it would terminate the wavelengths before they have a chance to reach the problematic high-energy (ultraviolet) wavelengths. Landau (1955) appears to be the first to have suggested this idea based on his more general desire to achieve something beyond quantum field theory, since Landau had a deep distrust of quantum field theory. More specifically, Landau’s field theoretic investigations revealed that even given renormalization, the short-distance behaviour continued to generate infinities (“Landau ghosts”) on account of the approach to the infinite bare masses and couplings (past the “screening” effect that Landau had himself discovered).[4] Pauli also makes several comments to this effect,[5] including the following remarks in a letter addressed to Abrikosov, Khalatnikov, and Pomeranchuk (in which he assigns priority of the idea to Landau):

I was very interested in Landau’s remarks on the possibility of a connection of the cut-off moment of quantum electrodynamics with gravitational interaction (his article “on quantum theory of fields” in the Bohr-festival volume). It appeals [sic] to me, that the situation regarding divergencies would be fundamentally changed, as soon as the light-cone itself is not any longer a c-number equation. Then every given direction in space-time would have some “probability to be on the light-cone”, which would be different from zero for a small but finite domain of directions. I doubt, however, that the conventional quantization of the g_{\mu \nu }-field is consistent under this circumstances. (Zürich, 15 August 1955; in Meyenn (2001, 329))

Given his obvious expertise in both general relativity and quantum field theory, one wonders why he didn’t do more work in the area of quantum gravity. Though as he suggests, it is clear that novel (i.e. unconventional) approaches are probably required (a far cry from his optimistic remarks in the 1929 paper with Heisenberg in which he claims that the general relativistic case would be much the same as electromagnetism). He was, nonetheless, certainly preoccupied with general relativity towards the end of his life—perhaps this “later-life” preoccupation with the unification of quantum and gravity (also shared by Eddington and Schrödinger) contributed in some small way to the poor reputation of the field at this stage.

Developing his earlier remarks to Abrikosov et al. (in the discussion after Klein’s talk), Pauli mentions Landau’s argument that for large cutoff momentum P, the gravitational coupling between a pair of electrons is of the same magnitude as the Coulomb forces. He notes that Landau’s relation G P^{2} \sim 1 is the same as Klein’s P \sim \hbar / l_{0} (where l_{0} = \sqrt{G \hbar c}). He writes:

[T]he connection [...] of the mathematical limitation of quantum electrodynamics with gravitation, pointed out by LANDAU and KLEIN, seems to me to hint at the indeterminacy in space-time of the light-cone, which is governed by probability laws in a quantized field theory, invariant with respect to the wider group of general relativity. It is possible that this new situation so different from quantized theories, invariant with respect to the LORENTZ group only, may help to overcome the divergence difficulties which are so intimately connected with a c-number equation for the light-cone in the latter theories (Pauli’s comments after Klein’s talk, in Mercier and Kervaire 1956, 69).

Pauli’s thoughts were borne out in one way (the light cone structure is affected as he suspects); however, it leaves a challenge behind in dealing with new divergences. As Bryce DeWitt pointed out, in Louis Witten’s important collection, Gravitation: An Introduction to Current Research, from 1962 (less than seven years after Pauli’s remarks),

[I]t must constantly be borne in mind that the “bad” divergences of quantum gravidynamics are of an essentially different kind from those of other field theories. They are direct consequences of the fact that the light cone itself gets shifted by the non-linearities of the theory. But the light-cone shift is precisely what gives the theory its unique interest, and a special effort should be made to separate the divergences which it generates from other diverg’ences. (DeWitt 1962, 374).

In the United States, the first PhD thesis to be written on quantum gravity was that of Bryce DeWitt, under the supervision of Julian Schwinger (at Harvard University, completed in 1949). DeWitt sought to revisit Rosenfeld’s work on the computation of gravitational self-energies (cf. Deser 1957). DeWitt would also revisit this idea of Landau’s that gravity might act as a natural regulator (DeWitt 1964). Though Landau didn’t explicitly mention the Planck scale (he placed the location of the cutoff much higher), Pauli clearly appeared to think that Landau had quantum gravitational effects in mind (or that he ought to have). It is clear that if there is a “fundamental length,” below which ordinary quantum field theoretic processes cannot operate, then one has what Landau sought. DeWitt was able to confirm that (at lowest order of perturbation) when gravity is included, the self-energies of charged particles (and the gravitons themselves) remain finite (though often very large). Here again, as in earlier parts, we see a link between minimum length scales and the notion of limits and domains of applicability of theory and concepts. The question of whether there is a physical cutoff naturally has theoretical links with programmes concerning the existence of a fundamental length, and discrete space(time) in general.[6]

More indirect, however, was Peter Bergmann’s method of utilising the fact that the gravitational field equations determined particle trajectories free of any notions of divergences. He believed this would follow from the analysis of Einstein, Hoffmann and Infeld, according to which the assumption of geodesy for a free particle’s motion was redundant, since it already could be seen to follow (by a method of successive approximation) from the field equations alone.[7]

Developing the cutoff idea, and the idea that there might be a minimal (fundamental) length, leads one quite naturally into the idea that space and time might not be continuous, but better modelled instead by a discrete lattice or similar structure. In the early days the cutoff was implemented in the kinematical structure, rather than having it emerge dynamically—whether the cutoff (discreteness) is fundamental or not is a different issue. This was suggested by several people. In a paper from 1930 Ambarzumian and Iwanenko (Chapter 25) argued for the introduction of a spatial lattice structure for physical space as a way of eliminating the infinite divergences from the self-energy of the electron. The basic idea was that the existence of a minimal length would imply a maximal frequency (p. 416). Alfred Schild (Chapter 30) investigated the properties of such a discrete lattice in order to see if it would break essential symmetries. In particular, he was responding to the objection that discrete theories would violate Lorentz invariance, which could manifest experimentally resulting in inconsistencies with known results.[8] He wasn’t able to devise a model to preserve all such symmetries, but enough to provide a plausible candidate for a background for a physical theory. Here again we find constraints operating on the various approaches to provide some sort of mechanism for the rejection and selection of theories or approaches—in this case the Lorentz symmetry of the classical theory.

Another discrete approach, of David van Dantzig (1938; 1956), was motivated by a combination of general covariance (as expressed in Einstein’s “point-coincidence” argument) and the definition of observability in such a theory. He argued that in a generally covariant theory the observable things will be coincidences: events (not shuffled by diffeomorphisms). Van Dantzig argues that in order to not introduce unmeasurable structure into the interpretation or formulation of one’s theory, one should dispense with the existence of a four-dimensional continuum, in favour of a discrete manifold of events. Peter Bergmann describes one such approach as one of “constructing “spaces” that have certain topological properties similar to those of point spaces in the large but do not possess “points” as elementary constituents” (Bergmann, following a talk of Wigner’s: Wigner (1956, 226)). The general approach lives on in several of the current approaches, including causal set theory and dynamical triangulations—though the conceptual basis (especially observability through invariance is absent from the latter case). Bergmann’s comments also draw attention to the “emergence” of continuous spacetime from a discrete structure (a problem at the root of causal set theory, though one in which progress has been made: see, e.g., Major, Rideout, and Surya (2007)).

There was nothing corresponding to paradigms in the early work. Nobody pursued a single programme for long enough—though Bergmann’s initial canonical quantization approach spawned a genuine research programme (along with a family of characteristic questions, having to do with “true observables” and the like) that has persisted. However, in the present day we do have a situation of what seem to be competing, coexisting paradigms (with elements of this sourcebook’s papers as ingredients). We can find the seeds of this landscape in the emergence of various “schools of research”, each tackling the problem of quantum gravity in a unique way. Often these schools themselves had seeds in the distinct tools that the researchers brought from their training, as physicists and mathematicians (recall that before the 1960s, it was rare to find general relativity taught outside of mathematics departments).

The idea at the focus of the papers in this part, that a discretization of space might go some way towards resolving the problems of short-distance physics, is of course rather natural and almost obvious. However, the initial developments were not linked to gravitational physics, although many of the results originally couched in non-gravitational work were carried over into that area. It was eventually realised, for example, that gravitation itself might be able to provide a physical foundation for discrete space and that given the dual nature of the metric field, quantum gravity should lead one to expect a discrete spacetime. Given this, the various results pursued independently of the quantum gravity problem (violation of Lorentz invariance and so on), become directly relevant.

There are three motivations underlying the notion of discrete space(time) in the early work:[9]

An ad hoc discretization using a lattice structure—often used as an approximation, for which the continuum limit would be taken later on.

An operational discretization using fundamental measurement limitations imposed by the uncertainty relations.

A discretization using a physical cutoff imposed (e.g. by gravity).

The first steps towards a field theory over a discrete space—along the first motivation (in the context of field theory)—were taken by Ambarzumian and Iwanenko in 1930. This paper also includes a discussion of whether time would need to be quantized, along with space, as a corollary. The argument is simple: a minimum length implies a maximum frequency which implies a minimum time interval \Delta t = \frac{1}{c}\Delta x.[10] They are concerned solely with the infinite self-energies that arise from the point-like nature of electrons. As they note, there seem to be two broad ways out of the predicament: give the electrons a finite size, or else restrict the spatial resolution to which one can probe (placing a limit on the validity of the theory—motivation two above). Since the former was thought to be not possible in quantum mechanics, they opt for the latter strategy. They resolve this “problem of space” by introducing a cubic lattice with grid points separated by some constant factor, a, to be determined (such that ordinary quantum theory is recovered as a\rightarrow 0). Differential equations are then replaced by discrete, difference equations.

This was followed by Heisenberg,[11] Ruark, March and several others, including, in England, Henry Flint. Flint was an interesting case, since he had his eye on the problem of unification of relativity and quantum mechanics in his work on fundamental length (via “ultimate measurements”—again, corresponding to the second motivation).[12] The Ambarzumian and Iwanenko paper was also directly cited by Schild, in his paper on discrete spacetime (included in this volume).

There were some other interesting attempts for “quantizing space” in the 30s. The most interesting is perhaps John Von Neumann’s (unpublished) proposal from 1937.[13] Von Neumann distinguishes two kinds of singularity: the point-particle singularity and the infinite degree of freedom singularity (resulting from the infinite number of parameters needed to describe a field). In a letter to Rudolf Ortvay from 1938 he describes his model for discrete spacetime as follows:

(1)

The x, y, z coordinates and the t are non-commuting operators.

(2)

The order of magnitude of commutators is \frac{h}{mc}. (That is to say, this is the uncertainty associated with a simultaneous measurement of coordinates.)

(3)

The whole structure has the Lorentz-symmetry.

(4)

Each of the x, y, z coordinates has a discrete spectrum: \pm 1/2, \pm 3/2, ...

(5)

The spectrum of the time t is continuous, from -\infty to +\infty.

(6)

When 4. and 5. are combined with 3. this comes out:

Given four real numbers \alpha, \beta, \gamma, \delta, the spectrum of the operator \alpha x + \beta y + \gamma z + \delta t is as follows:

(a)

If \alpha ^{2} + \beta ^{2} + \gamma ^{2} - \delta ^{2} > 0 then it is discrete: \pm \epsilon /2, \pm 3 \epsilon /2, ... ,

where \epsilon = \sqrt{\alpha ^{2} + \beta ^{2} + \gamma ^{2} - \delta ^{2}}.

(b)

If \alpha ^{2} + \beta ^{2} + \gamma ^{2} - \delta ^{2} < 0 (indeed even when =0) then it is continuous from -\infty to +\infty.

So this a “discrete” crystalline space with “continuous” time, which has not only spherical symmetry (even though it is a “crystal”!), but is even invariant with respect to changes of the reference system given by Lorentz transformations, and so shows the proper Lorentz-FitzGerald contraction phenomena. (This is made possible, of course, by the non-commuting nature of the coordinates.) (von Neumann, letter to Rudolf Ortvay: March 17, 1938 [Rédei 2005, 22])

In a letter to Dirac (dated January 27, 1934) he writes:

It should be perhaps desirable, to have operators X, Y, Z which gave discrete (point) spectra, in order to avoid the difficulties connected with the point electron (in electrodynamics). (Rédei 2005, 21)

Dirac replied the following month (February 28, 1934) pointing out that the model was not invariant under displacement of the origin of the reference frame defined by the X, Y, Z.

Quantization here is viewed, then, as a cutoff to prevent the ability to resolve to point-like distances. The problem with such accounts is that they are physically ad hoc (motivation one from above). Von Neumann did not pursue the idea further for this reason: “because [he] considered it very artificial and arbitrary” (Rédei 2005, 22).

Heisenberg was inspired primarily by the second motivation, though it mixed with the first, in order to tame the infinite self-energy of electrons.[14] His first thoughts about discretisation can be found in a long letter to Bohr from March 1930 (translated into English in Carazza and Kragh (1995), along with a reconstruction of the logic of the argument it contains)—one wonders whether he was aware of Ambarzumian’s and Iwanenko’s work, which is remarkably similar (as mentioned, Ambarzumian and Iwanenko note, at the proofs stage of their paper, that they were aware of Heisenberg’s attempt, though it is hard to discern whether their work was initially written without knowledge of this). The idea is also to divide space up into a cubic lattice, where the cells have volume r_{0}^{3} = (h/Mc)^{3}. The length \root 3 \of {r_{0}} (the electron radius) was then the “elementary length”. He called the world described by this theory “gitterwelt” (“lattice world”). The self-energy of an electron would be rendered finite in the gitterwelt—a point Heisenberg returned to in his paper “Die Selbstenergie des Elektrons” (submitted in August of that year). As Heisenberg also notes, in the given scheme differential equations would have to be replaced with difference equations.[15] A central problem, as Heisenberg saw it (and as would deter others from the discrete space idea) was that relativistic invariance was spoiled by any scheme that introduced a fundamental length—this assumption was progressively taken apart in papers from the late 1930s onwards. (Heisenberg also pointed to difficulties in making the space isotropic; as well as with energy, momentum, and charge conservation: for these reasons he asked Bohr whether he thought the idea “completely mad”!) But beyond this breakdown of Lorentz invariance, the other target of Heisenberg’s 1930 paper was to show that there are wider problems with field theory that go beyond the problem of infinite self-energy—this became part of a general programme of getting clearer on the distinct kinds of divergences in physical theories (on which, see the introduction to the previous part of this volume).

In his 1938 paper (Chapter 26: “Über die in der Theorie der Elementarteilchen auftretende universelle Länge”), Heisenberg explicitly ignores gravitational interactions “which hardly play a role in nuclear physics,” focusing on \hbar and c alone. This is part of a general to and fro with respect to the role of gravitation in elementary particle physics. However, when discussing the “universal length” he does briefly return to the issue, though again to dismiss gravity’s role in the fundamental length. As is standard, he considers the electromagnetic analogy, comparing the gravitational interaction of photons with the electrical interaction of electrons. But he notes a crucial dis-analogy: introducing the gravitational constant (Heisenberg uses \gamma rather than G) together with \hbar and c can be combined to generate the (Planck) length: l = \sqrt{\hbar \gamma / c^{3}} (which Heisenberg computes to be 4 \times 10^{-33}cm). However, given the vast distances separating these domains, Heisenberg points out that the problems ass’ociated with his r_{0} (the electron radius) ought to be resolved first, as the most urgent task. In other words, there is a practical argument here for the neglect of issues having to do with quantum gravity.

The reason for this urgency were the difficulties faced by Fermi’s theory of \beta decay, based on Pauli’s neutrino hypothesis, which was found to suffer from divergences of an extreme (i.e. unrenormalizable in modern parlance) kind—involving the divergence of (Born approximation) cross sections as the energy of the incident particles went to infinity—so that the perturbation technique for treating interactions didn’t give sensible answers.[16] Of course, we know that this problem was pointing to a limit with the then current quantum field theory. But Heisenberg, viewing Fermi’s theory as a fundamental (and unified, in terms of weak and strong forces, with a single coupling constant) theory, took it to point to another source in which one could only resolve distances to certain distances, again close to his r_{0}.[17] In this case Heisenberg drew attention to the particle multiplicity (“explosionen”) in cosmic ray showers in which many particles are created: the particle production would limit the resolution (so that r_{0} represents a fundamental limit in this sense: physics becomes “turbulent” at shorter lengths as the coupling blows up). Of course, there was a limit, but the limit was theoretical rather than practical: there was a layer of particle physics below that captured by Fermi’s theory. The short-distance, strong interaction physics that followed this was a major impetus to quantum gravity physics since gravitational and strong interactions had similar non-linearities (due to the self-interacting nature of the forces)—though, of course, gravity is universal (couples to all sources of energy equally). Hence, a new analogy between these forces, and to a lesser extent with electrodynamics, took hold.[18]

We have seen that Bronstein had already written on related issues in 1936, pointing out that there exist quantum measurement restrictions beyond the commutation relations in the case of gravitational measurements, since there cannot be bodies of arbitrarily large mass density (cf. Gorelik and Frenkel 1994, 106). Bronstein thought this called for a revision of spacetime concepts (as did Heisenberg, though for different reasons). Unfortunately, his untimely death means that we don’t know how or if he intended to pursue this revision.

The third motivation was discussed, mostly informally, in our pre-1950 period, by Pauli and Landau amongst others, but the idea was not fully developed in published form beyond mere suggestions.

Hartland Snyder’s is probably the best known early work on discrete spacetime—the paper is truly a citation classic, with 1714 citations at the time of writing.[19] In this case the resulting spacetime is explicitly presented as quantized, with the spacetime coordinates themselves represented by Hermitian operators with discrete spectra. We have already seen this basic idea, of course, with Heisenberg and von Neumann. The innovation is to properly formalise the idea and establish that the discrete space idea need not violate Lorentz invariance.

Snyder (1947) returned to the subject the following year, with a paper applying the quantized spacetime concept to the electromagnetic field. There is then a trail leading from Snyder to Schild, in which the mathematical properties of quantized spacetime are uncovered.[20]

C. N. Yang (Chapter 28) tackled a serious issue with Snyder’s model, namely that it violates translation invariance whenever the coordinates are not a continuum. However, a continuum clashes with the fundamental (i.e. non-epistemic) minimum length of the model.[21] Yang resolves the translation issue, but a problem of scale persists, namely in the form of a curvature of the universe at odds with what we observe (curvatures are of the order of the Planck scale rather than the Hubble radius).

It is rather interesting that Alfred Schild published his work on discrete spacetime around the same time that he transitioned into research on the canonical quantization of general relativity, following Dirac’s influential 1949 lectures at the International Mathematical Congress in Canada, which Schild attended with his Masters student Felix Pirani.[22] Yet there is no mention of gravity in his paper on discrete spacetime, despite the fact that together with Pirani, Schild explicitly quantised the spacetime metric. This clearly reveals (perhaps rather surprisingly) that the project of “quantization of gravity” had not yet been linked to what we now call “quantum spacetime.” The focus is instead on the construction of a discrete model of spacetime that is as close as possible to Lorentz invariant, and the context is the problematic divergences of standard quantum field theory. Schild’s basic object is a hypercubic lattice, with time coordinate included amongst the spatial coordinates. He deals with ( c=1) Lorentz transformations that map a 3-lattice onto itself (where the 3-lattice takes on integer coordinates).

While Snyder’s approach was indeed Lorentz invariant, it made use of the rather awkward idea that spacetime coordinates were non-commuting operators (so that spacetime functions become Hilbert space operators) and was not translation-invariant. Schild uses coordinates that are integer multiples of a fundamental length (rather than having eigenvalues that are integer multiples as with Synder), and so more along the lines of the proposals of Ambarzumian and Iwanenko et al.. Schild’s goal is likewise to show that a common objection against discrete approaches to eliminating the divergences—that they violate Lorentz invariance due to the frame dependence of the “minimum” cell size—is only partially correct since one can construct models that are invariant under a large subgroup (the discrete subgroup) of the Lorentz group. These, he suggests are in fact physically viable (unlike Snyder’s and Yang’s), and cast in a model closer to ordinary spacetime, thus undermining a host of common objections and making discrete models in principle a genuine possibility for fundamental physical theory—though, as he admits, his own model suffers from physical inconsistencies to do with a radically oversized minimal velocity.[23]

Nathan Rosen (Chapter 29) introduces statistical considerations into the treatment of a discrete space: his elementary volumes are related to position measurement uncertainties (that is, to practical limitations: no infinitesimal measurement rods, therefore no physical point-like measurements). More specifically, the measurement of spatial coordinates of elementary particles (electrons) introduces inaccuracy into the measurement results such that repeated measurements will generate values sitting around the mean of a Gaussian distribution. His aim is, as with other proposals we’ve considered, to eliminate singularities (relating to the second motivation again). The resulting picture is not so very different from the Synderian one of a non-commutative space. However, the discreteness here is epistemological, coming from the difficulties involved in pinning down a spacetime point.

There is a very (later) Eddingtonian quality to this, especially the splitting of the abstract space from the observable space, which corresponds to Eddington’s geometrical and physical frames.[24] Volumes in the observable space correspond to points in the abstract space. Lorentz invariance is preserved in this scheme only in the abstract space; yet Rosen suggests that a kind of translation manual could be established between transformations in this space and real physical transformations in the observable space.

There is also an interesting parallel here to some of the issues regarding the “reality of spacetime points” (e.g. in the context of the hole argument in general relativity). Rosen argues that the value of a physical quantity at a point is not directly observable, so that physical laws should not be based on such quantities. What is not clear is whether, according to Rosen, the world (ontology) tracks epistemology so that our laws must be written this way because the world is that way so that only the mean values of quantities over volumes have any physical meaning at all.

Rosen reviewed a closely related paper by Averbah and Medvedev in 1949. He also later returned to a similar idea, writing with Asher Peres, in 1960, though this time explicitly linking to measurement of the gravitational field. By this stage they viewed the existence of quantum uncertainties in these measurements (in the mean values of the Christoffel symbols) as pointing to the necessity of quantizing the gravitational field. Though we don’t see any explicit discussion of the “discrete space-gravitation” connection, the works presented here nonetheless contain crucial evolutionary steps. The recognition that playing around with the structure of space(time) might offer up cures for some of the difficulties of quantum field theory was an early one; linking this up with the way in which general relativity includes the geometrical structure of spacetime as one of the dynamical variables took somewhat longer.

References

Brown, Laurie M. (1993). Renormalization: From Lorentz to Landau (and Beyond). Berlin: Springer.

Brown, Laurie M., and Helmut Rechenberg (1996). The Origin of the Concept of Nuclear Forces. Bristol; Philadelphia: Institute of Physics Publishing.

Carazza, Bruno, and Helge Kragh (1995). “Heisenberg’s Lattice World: The 1930 Theory Sketch.” American Journal of Physics 63 (7): 595–605.

Cushing, James T. (1990). Theory Construction and Selection in Modern Physics: The S Matrix. Cambridge: Cambridge University Press.

Dantzig, David van (1938). “Some Possibilities for the Future Development of the Notions of Space and Time.” Erkenntnis 7: 142–46.

——— (1956). “On the Relation Between Geometry and Physics and the Concept of Space-Time.” In Jubilee of Relativity Theory: Proceedings, edited by André Mercier and Michel Kervaire, 48–53. Basel: Birkhäuser.

Deser, Stanley (1957). “General Relativity and the Divergence Problem in Quantum Field Theory.” Reviews of Modern Physics 29 (3): 417–23.

DeWitt, Bryce (1962). “The Quantization of Geometry.” In Gravitation: An Introduction to Current Research, edited by Louis Witten, 266–381. New York: Wiley.

——— (1964). “Gravity: A Universal Regulator?” Physical Review Letters 13 (3): 114–18.

Gorelik, Gennady E., and Victor Ya. Frenkel (1994). Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties. Basel: Birkhäuser.

Hagar, Amit (2014). Discrete or Continuous? The Quest for Fundamental Length in Modern Physics. Cambridge: Cambridge University Press.

Ivanenko, Dimitri (1931). “Die Beobachtbarkeit in Der Diracschen Theorie.” Zeitschrift Für Physik 72 (9–10): 621–24.

Klein, Oskar (1956). “Generalisations of Einstein’s Theory of Gravitation Considered from the Point of View of Quantum Field Theory.” In Jubilee of Relativity Theory: Proceedings, edited by André Mercier and Michel Kervaire, 58–71. Basel: Birkhäuser.

Kragh, Helge, and Bruno Carazza (1994). “From Time Atoms to Space-Time Quantization: The Idea of Discrete Time, ca 1925–1936.” Studies in History and Philosophy of Science 25 (3): 437–62.

Landau, Lev D. (1955). “On the Quantum Theory of Fields.” In Niels Bohr and the Development of Physics, edited by Wolfgang Pauli. London: Pergamon Press.

Major, Seth, David Rideout, and Sumati Surya (2007). “On Recovering Continuum Topology from a Causal Set.” Journal of Mathematical Physics 48 (032501).

Mercier, André, and Michel Kervaire, eds. (1956). Jubilee of Relativity Theory: Proceedings. Basel: Birkhäuser.

Meyenn, Karl von, ed. (2001). Wolfgang Pauli: Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.a. Vols. Vol. 4, Part 3: 1955–1956. Berlin: Springer.

Oppenheimer, J. Robert, and Hartland Snyder (1939). “On Continued Gravitational Contraction.” Physical Review 56: 455–59.

Rédei, Miklós, ed. (2005). John von Neumann: Selected Letters. Providence: American Mathematical Society.

Rickles, Dean (2014). A Brief History of String Theory: From Dual Models to M-Theory. Berlin / Heidelberg: Springer.

Rovelli, Carlo, and Simone Speziale (2003). “Reconcile Planck-Scale Discreteness and the Lorentz-Fitzgerald Contraction.” Physical Review D 67 (064019).

Rueger, Alexander (1992). “Attitudes Towards Infinities: Responses to Anomalies in Quantum Electrodynamics, 1927–1947.” Historical Studies in the Physical and Biological Sciences 22: 309–37.

Snyder, Hartland (1947). “The Electromagnetic Field in Quantized Space-Time.” Physical Review D 72: 68.

Wess, Julius (2001). “Nonabelian Gauge Theories on Noncommutative Spaces.” Communications in Mathematical Physics 219 (1): 247–57.

Wheeler, John A. (1968). “Superspace and the Nature of Quantum Geometrodynamics.” In Battelle Rencontres: 1967 Lectures in Mathematics and Physics, edited by Cécile DeWitt-Morette and John A. Wheeler, 242–307. New York / Amsterdam: W. A. Benjamin.

Wigner, Eugene P. (1956). “Relativistic Invariance of the Quantum Mechanical Equation.” In Jubilee of Relativity Theory: Proceedings, edited by André Mercier and Michel Kervaire, 210–26. Basel: Birkhäuser.

Footnotes

[1] A good philosophical treatment of the reactions to these divergences is Rueger (1992).

[2] The alternative, to be found in superstring theory, is to postulate a finite size for the elementary entities (in the earlier period, this would have been, e.g., the electron radius).

[3] What the non-linear analysis eventually revealed was that gravity’s divergences were more complex than those of electrodynamics, since they involved a shifting of the light cone so that light cone singularities would be “smoothed out” by fluctuations in a quantum theory of gravity (a nice result of the quantum geometry inherent in quantization of the metric): Klein 1956; Landau 1955; Deser 1957. We return to this briefly below.

[4] Later work would reveal that such Landau ghosts could be dealt with in the context of renormalization group theory, but at the time it appeared as though quantum field theory was suffering from an incurable illness—for more on this, see Brown (1993, 21).

[5] See also Pauli’s letter to Källén (dated 24th April, 1955: pp. 207–208); Peierls’ letter to Pauli (dated 9th May, 1955: pp. 228–229); Pauli’s letter to van Hove (dated 11th May, 1955: 230–231); Heisenberg’s letter to Pauli (dated 11th, May 1955: pp. 234–235).

[6] As DeWitt puts it: “The dimension 10^{-32}cm constitutes a fundamental limit on the smallness of allowable measurement domains. Below this limit it is impossible to interpret the results of measurement in terms of properties or states characterising individual systems under observation” (DeWitt 1962, 373).

[7] I might also note here that ultimately string theory emerged from the divergences problems facing quantum field theories of fields other than the electromagnetic field (particularly the strong interaction). In particular, since the perturbative approach breaks down when the coupling constant determining the strength of an interaction is large (as in strong interaction physics), alternative approaches were sought in the late 1950s and throughout the 1960s. One of the more popular of these approaches combined Heisenberg’s S-matrix theory with dispersion theory. The S-matrix is a tool to encode all possible collision processes. Heisenberg suggested that one take this to embody what was relevant about the physics of collision processes. In particular, all that was observable were the inputs and outputs of collision processes, observed when the particles are far enough apart in spacetime to be non-interacting, or free. This black box approach to physics was very much inspired by the Copenhagen philosophy. The dispersion relation approach to physics tried to construct physical theories on the basis of a few central physical axioms, such as unitarity (conservation of probabilities), Lorentz invariance, and causality (effects can’t precede causes). These two approaches were combined, by Geoff Chew amongst others, so that the focus was on the analytic properties of the S-matrix. One model for the S-matrix, incorporating some other principles thought to be involved in strong interaction physics, was the Veneziano model. This used the Euler beta function to encode the various desirable properties of the S-matrix. The model was found to be generated by a dynamical theory of strings. (See Cushing (1990) for a detailed historico-philosophical account of the early development of string theory, or Rickles (2014) for a more recent account.)

[8] This same objection to discrete models surfaces again in present-day discussions of discrete space in quantum gravity (a fairly generic prediction of several approaches), especially in the context of loop quantum gravity which directly predicts (at least at the kinematic level) geometrical operators with a discrete spectrum. Given that there is supposed to be a fundamental length (namely the Planck length, and corresponding fundamental times and masses) in these approaches, it makes sense to ask if observers in relative motion will agree on this length: why no Lorentz-FitzGerald contraction for boosted observers, rendering the notion of a minimum length incoherent? Why is a length measurement for the minimum length case not subject to the usual frame dependence? According to Carlo Rovelli (one of the primary architects of loop quantum gravity that itself appears to face the problem) and Simone Speziale, quantum mechanics is the key to avoiding this “discreteness/invariance” conflict: “the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable [so that the] boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area” (Rovelli and Speziale 2003, 064019). They elaborate as follows, linking directly with issues of quantum spacetime: “The geometry of space comes from a quantum field, the quantum gravitational field. Therefore the observable properties of the geometry, such as, in particular, a length, or an area, are observable properties of a quantum physical system. A measurement of a length is therefore a measurement in the quantum mechanical sense. Generically, quantum theory does not predict an observable value: it predicts a probability distribution of possible observable values. Given a surface moving in spacetime, the two measurements of its area performed by two observers O and O^\prime boosted with respect to one another are two entirely distinct quantum measurements. Correspondingly, in the theory there are two distinct operators A and A^\prime, associated to these two measurements. Now, our main point is the technical observation that A and A^\prime do not commute: [A,A^\prime ] ̸= 0. This is because A and A^\prime depend on the gravitational field on two distinct 2d surfaces in spacetime [...] and a field operator does not commute with itself at different times”. Hagar (2014, §8.4.4) contains a useful, detailed discussion of this problem.

[9] As Rueger makes clear (Rueger 1992, 317), prior to the 1930s there was a sense that the infinities were simply a hangover from the classical theory that if cured first (classically) would not reassert themselves at the quantum level. This was not the case, and it became clear that there existed specifically quantum divergences.

[10] In another paper from the following year, Iwanenko reiterates that the value \lambda \sim \frac{h}{mc} also determines a “chronon”: “Dieser Wert hat schon als kleinste definierbare Entfernung zu gelten und nicht der Elektronenradins. Mit der kleinsten Entfernung hängt die kleinste Zeitspanne zusammen” (Ivanenko 1931, 623). As Kragh and Carazza note, there were earlier speculations, with similar results, about time atoms from Pokrowski and Fürth (Kragh and Carazza 1994, 457–458). Indeed, they show that the 20s and 30s were positively teeming with discrete space, time, and spacetime proposals. However, many of them are detached from the central problems of field theories that concern us here.

[11] In fact, in the acknowledgements to their paper, Ambarzumian and Iwanenko refer to analogous work of Heisenberg that they had only just became aware of at the time of publication.

[12] In his notebook (from 1950) there is a section on “The Theory of Relativity and the Quantum Theory” in which he nails down his project: “The underlying theme of this work is the union which exists between the theory of relativity and the quantum theory and the purpose is to portray it by means of geometry and a theory of measurement” (Henry Flint Papers, University of London, Document B53: p. 1).

[13] The manuscript is entitled “Quantum Mechanics of Infinite Systems” see Rédei (2005, 21–22).

[14] Interestingly, Heisenberg had already briefly considered the idea of letting spatial coordinates be non-commuting in 1930 in order to generate a minimum length from uncertainty relations. He put this idea to Rudolf Peierls asking for any suggestions, including any input from Pauli. Julius Wess (2001, 1) claims that Heisenberg relayed it to Peierls (his student), who relayed the idea to Pauli who relayed it to Oppenheimer (his student), who relayed it to Hartland Snyder (his student: see below)! This occurs over a period of 15–16 years.

[15] Carazza and Kragh (1995) argue that Heisenberg did not really endorse a discrete space at this stage, but rather used discreteness only at the level of derivatives with respect to spatial coordinates (which are indeed replaced by discrete, finite differences).

[16] The is the famous four-fermion coupling G_{F} which was not properly understood until the electroweak theory was developed, and the machinery of gauge theory was applied, along with Yukawa’s idea of mediation by a new kind of boson (the “U-quantum” or mesotron) which replaced the four-fermion term. The evidence for mesons came in 1937, when they were isolated in cosmic rays. It should perhaps also be said that the realisation that all would not be plain sailing with respect to the other forces of nature shifted the focus onto the peculiarities of gravitation. See Chapters 3 and 4 of Brown and Rechenberg (1996) for a historical study of the Fermi-field theory.

[17] Now the length involves the mesotron mass \mu, \hbar /\mu c, derived from Yukawa’s theory. Note that Heisenberg’s persistent belief in a fundamental, universal length can be seen as more reasonable given that there is a remarkable coincidence between the electron radius and this meson mass (and so the range of the nuclear forces).

[18] As Brown and Rechenberg make clear, the existence of cosmic ray phenomena was pivotal precisely in that it served to delineate the borders of the known physical theories, pointing out exactly when they would break down, see Brown and Rechenberg (1996, 72). Heisenberg was, of course, wrong in thinking that Fermi’s theory was fundamental: there was new physics that Heisenberg was not then privy to.

[19] Of course, this is the same Snyder who had worked with Robert Oppenheimer, in 1939, on the fate of very massive collapsing stars (approximated by an homogeneous, zero pressure ball of dust), showing that a one-way membrane (an “event horizon” in modern parlance) would emerge from the process and that a final singularity would also result—Landau had earlier noted the existence of a critical mass in 1932, and Chandrasekhar had shown in 1931 that the electron degeneracy pressure could not withstand further collapse for stars greater than 1.3 solar masses. It is rather odd that Snyder never made any link between these two streams of his work—continued collapse to a singularity and discrete space—since the former involves the reduction of a system’s dimensions to values small enough (perhaps indefinitely small) to be relevant for the latter. (Oppenheimer and Snyder even write, “Physically such a singularity would mean that the expression used for the energy-momentum tensor does not take account of some essential physical fact which would really smooth the singularity out”, Oppenheimer and Snyder (1939, 456). Later, John Wheeler would bring the two together via the Planck length: two areas where the “dynamics of geometry” fails to lend itself to classical analysis (Wheeler 1968, 253–254). This work would lead, ultimately, to Wheeler’s notion of “spacetime foam.”

[20] Bergmann and Brunings briefly refer to Snyder, if only to distance their quantised metric variables from his: their coordinates, as they say, “commute with each other, but not with the energy-momentum densities”. They continue: “The dynamical character of any particle coordinates follows automatically, but probably does not exhaust the physical significance of the coordinate commutation relations” (see Chapter 33). This highlights the continuity, at least, between Snyder’s (and the other related) work on quantized/discrete spaces and quantum gravity research.

[21] We saw above, in footnote 8, how Rovelli and Speziale manage to sidestep the problem by introducing probabilities for measurement outcomes.

[22] Indeed, Schild’s paper appears in the very same journal as Dirac’s paper, in the issue directly preceding that containing the paper that would inspire Schild’s work on the quantisation of the gravitational field.

[23] This shortcoming was partially eliminated by E. L. Hill in 1955 by restricting the values of spacetime variables to rational numbers—partially, because the resulting space does not quite live up to the “discrete” moniker. As Hill notes in a footnote in this paper, his Master’s student, C. N. Kelber was working on this same problem of Lorentz invariance violation at the same time as Schild. There is some correspondence between Schild and Kelber, where the latter explains that he has a model that involves non-homogeneous Lorentz transformations so that the origin is not fixed for all observers (Kelber, letter to Schild, June 21st 1948).

[24] Rosen adds a statement to this effect, though pointing out that he was unaware of Eddington’s work at the time of writing. Interestingly, this information was relayed to Rosen by M. F. M. Osborne, more famous now perhaps as an early econophysicist, but who also did early work on measurement restrictions, and minimal length, in quantum general relativity.

25. Victor Ambarzumian and Dmitri Iwanenko (1930): Zur Frage nach Vermeidung der unendlichen Selbstrückwirkung des Elektrons

Victor Ambarzumian and Dmitri Iwanenko (1930). Zur Frage nach Vermeidung der unendlichen Selbstrückwirkung des Elektrons. Zeitschrift für Physik, 64: 563–567.

26. Werner Heisenberg (1938): Über die in der Theorie der Elementarteilchen auftretende universelle Länge

Werner Heisenberg (1938). Über die in der Theorie der Elementarteilchen auftretende universelle Länge. Annalen der Physik, 424: 20–33.

27. Hartland S. Snyder (1946): Quantized Space-Time

Hartland S. Snyder (1946). Quantized Space-Time. Physical Review, 71: 38–41.

28. Chen-Ning Yang (1947): On Quantized Space-Time

Chen-Ning Yang (1947). On Quantized Space-Time. Physical Review, 72: 874.

29. Nathan Rosen (1947): Statistical Geometry and Fundamental Particles

Nathan Rosen (1947). Statistical Geometry and Fundamental Particles. Physical Review, 72: 298–303.

30. Alfred Schild (1948): Discrete Space-Time and Integral Lorentz Transformations

Alfred Schild (1948). Discrete Space-Time and Integral Lorentz Transformations. Physical Review, 73: 414–415.

Gauge and Constraints

31. The Genesis of Canonical Quantum Gravity

Alexander Blum and Donald Salisbury

The late 1940s saw the solution of the divergence difficulties of quantum field theory, or at least quantum electrodynamics.[1] The renormalization program was successful not only in removing the infinities, but also in giving precise values to the finite remainders. These were in excellent agreement with newly discovered precision effects, such as the Lamb shift and the anomalous magnetic moment of the electron. The renormalization techniques were, at least initially, based on the covariant quantization procedure discussed in Chapter 17 and their application to the quantization of gravity was taken up in the 1950s.

In parallel, and sometimes in direct opposition, to the development of renormalized quantum field theory (QFT), a second route to quantum gravity was developed in the late 1940s, initially by Peter Bergmann. Bergmann had been an assistant of Einstein from 1936 to 1941 and had worked with Einstein on the attempts to construct a five-dimensional unified field theory (UFT).[2] But unlike Einstein, Bergmann was not in radical opposition to modern quantum theory, having learned it as a student of Philipp Frank in Prague in the mid-1930s. After obtaining a permanent position at Syracuse University, Bergmann thus abandoned the pursuit of a unified field theory and set himself to reconcile general relativity (GR) and quantum theory through the construction of a quantum theory of gravity.

Bergmann was little concerned with the contemporary rapid development of renormalized quantum field theory. And indeed, the covariant QED of the day did not seem to harmonize well with his vision of quantum gravity. The QED of Schwinger was construed as the quantization of free, linear field theories, both for matter and the electromagnetic field, which were then perturbed by the introduction of interactions between the fields, leading to the non-trivial time evolution of the quantum state in the interaction picture.

But for Bergmann the essential element of GR was its non-linearity, because it made possible the derivation of the equations of motion of point particles directly from the field equations, as outlined by Einstein, Infeld and Hoffmann (EIH) during Bergmann’s time with Einstein. Bergmann’s hope was that this feature would carry over from the classical to the quantum theory, so that a quantum theory of gravity would directly lead to the quantum equations of motion of point particles as well. He was thus envisioning not merely an appropriation of the field theory of general relativity through quantum theory by the methods of field quantization, but rather a true marriage between general relativity and quantum theory, where the introduction of general relativity would help to solve the divergence difficulties of quantum field theory, which he, like many others, saw as originating from the difficulties of having point-like particles.

Of course this approach was not entirely up to date with the current state of quantum field theory, which had in a sense dispensed with the quantum mechanical notion of in principle localizable, point-like entities. Photons were certainly not perfectly localizable, nor were charged particles in electron-positron theory. And Bergmann had to admit that already classically the EIH procedure was only feasible if one disregarded splitting world lines, as in the emission of a photon or electron-positron pair creation. But these difficulties were put aside for the moment. Bergmann was after a quantum theory of point particles and the gravitational field.

The predilection for the EIH procedure was not the only inheritance from Bergmann’s time with Einstein. From his work on UFT, he also had a clear understanding that GR was not to be considered the final field theory of gravitation. In his program, he thus did not engage directly with GR and the quantization of the space-time metric, but preferred to discuss the quantization of a more general field theory that kept what he considered to be the essential elements of GR: general covariance and non-linearity, which together ensured the feasibility of the EIH-type determination of particle trajectories.

The study of the quantization of a field theory mainly characterized by its invariance properties unwittingly brought him very close to Rosenfeld’s first 1930 paper on what we have called the momentum difficulty. And indeed, in his first paper on the subject (Chapter 32), Bergmann showed not only that a general non-linear, generally covariant field theory would have all the attractive features of GR (EIH determination of the equations of motion, as well as energy conservation, at least in the same sense as in GR), but would also lead to identities involving the canonical momenta, of the type that Rosenfeld had first discovered. There are slight differences in the treatment, of course: Bergmann did not consider spinorial matter and thus did not introduce tetrads or local Lorentz symmetry.[3] Without tetrads, even for the case of regular general relativity, Bergmann could not have constructed a Lagrangian that both contained only first-order derivatives of the field (metric) quantities and was at the same time a scalar. He thus treated a more general case than Rosenfeld’s, allowing for the Lagrangian to change by a total divergence under a general coordinate transformation.[4] Bergmann’s derivation of the constraints was thus somewhat different from Rosenfeld’s and went via the (generalized) contracted Bianchi identities. This actually lent a certain degree of coherence to Bergmann’s overall approach, as the Bianchi identities also played an essential role in the generalized derivation of the EIH result.

Bergmann identified the identities (and the non-unique time development they implied) as the central challenge for the quantization of a generally covariant field theory. He was well aware that this problem could be remedied by the imposition of gauge, or rather, in this case, coordinate conditions, but general covariance was such a central concept to him that he instead aimed for constructing a Hamiltonian in which the arbitrariness of coordinate choice showed up explicitly, in the form of four arbitrary functions. He was thus aiming for a quantization procedure very alike that of Rosenfeld, yet without, it should be pointed out, being aware of Rosenfeld’s work.

The second paper he published in that same year, together with his colleague Johanna Brunings, represented a further step towards quantization (Chapter 33). The central point Bergmann wanted to address here was how to carry over the EIH method into quantum theory. The difficulty he had to address was the following: If there are singular worldlines in space-time, where the field equations are not satisfied, then one should not integrate over these singular regions in the action integral, whose minimization gives the field equations. However, without first solving the field equations, one cannot know the exact singular particle trajectories. Bergmann’s solution was the “parameter formalism.” The idea was that since the spacetime trajectories of the singularities could not be known in advance, a four-dimensional multiply connected parameter space (t, u^s) could be utilized in which parameter tubes could be excised without making a commitment as to the corresponding spacetime particle paths. The original spacetime coordinates x^{\rho } were taken to be functions of these parameters, thus becoming field variables on par with the gravitational field, dynamical equations of motion and all. The introduction of parameters was, as acknowledged by Bergmann, closely related to Weiss’s approach discussed in Chapter 17: For Weiss, too, the field variables were no longer functions of the space-time coordinates, but rather of the parameters defining a point on the hypersurface, as well as of a further parameter ( \lambda in Weiss’s notation, t in Bergmann’s), labelling a member of a sequence of hypersurfaces. And for Weiss the original space-time coordinates were also functions of the parameters. However, in Weiss’s case this was merely to be understood as the definition of a specific foliation of space-time. As Bergmann pointed out, the emergence of the original coordinates as dynamical variables on the same footing as the field variables y^{A}, which was the essential point for Bergmann, was not discussed by Weiss, who consequently also did not investigate the momenta canonically conjugate to the original coordinates.

Bergmann’s parameter formalism was now covariant not just under coordinate transformations but also under arbitrary reparameterizations. This covariance brought with it four new identities. Three of these (corresponding to the three spatial parameters u^s) could be straightforwardly deduced from the definition of the new canonical momenta conjugate to the now-dynamical spacetime coordinates. Repeating this procedure for the fourth (time) parameter t, did not directly deliver another constraint. Instead, Bergmann could show in this manner that the Lagrangian in the parameter formalism was homogeneous of degree one, in both the field velocities \dot{y}_A and the coordinate velocities \dot{x}^\rho, where the dot refers to differentiation with respect to parameter time tt in both cases. From this fact the existence of a further constraint equation g=0 could be deduced.

This function g had the properties of the Hamiltonian density, that is, it generated the canonical field equations of motion through functional differentiation. Consequently, using Poisson brackets (in the parameter version, first introduced by Weiss in the 1930s, but now, in Bergmann and Brunings’ approach, also including the space-time coordinates as dynamical variables) the spatial integral of g (the Hamiltonian) could be used to determine the time derivative of some functional of the canonical variables.[5] They then claimed to show, by calculating the Poisson bracket of the Hamiltonian with an arbitrary linear combination of the constraints, that all eight constraints had vanishing time derivatives, and that consequently no further constraints had to be considered. As Bergmann himself would soon realize, this proof was fallacious.[6] They also outlined how to proceed to a quantum theory, in the same manner proposed by Rosenfeld, that is, by turning the Poisson brackets into commutators and the constraints into auxiliary conditions on the wave function.

But although several important properties of the constraint g were known, Bergmann and Brunings did not provide a method for explicitly constructing it. They could show that it was not unique: For example, if one added to g some linear combination of the remaining seven constraints, the resulting function would of course still be a constraint and could also still be taken as a Hamiltonian density, the new canonical field equations of motion being related to the ones of the original g by a suitable canonical transformation of the dynamical variables along with a transformation of the space-time parameters. This implied that some specific choice of Hamiltonian density corresponded to choosing a coordinate condition and thus losing general covariance. But Bergmann and Brunings neither showed how to construct a specific Hamiltonian, corresponding to a particular coordinate condition, nor a general Hamiltonian involving arbitrary functions. They thus had reached, in 1949, about the same point that Rosenfeld had reached in 1930. The next step was clearly the explicit construction of a Hamiltonian, only that now, in Bergmann’s parameterized approach, this Hamiltonian would not just generate the field equations of motion and time evolution of physical variables, but would also itself be one of the constraints.[7]

In reaching this next step, Bergmann’s group was beaten by Felix Pirani and Alfred Schild. Their work built on an approach to constrained Hamiltonian dynamics quite different from that of Rosenfeld or Bergmann, presented by Paul Dirac in August/September 1949 at the Summer Seminar of the Canadian Mathematical Congress (Chapter 34), a newly established graduate-level summer school, which in that year (its second installment) had a special focus on mathematical physics, other lecturers including Homi J. Bhabha (on quantum field theory) and Laurent Schwartz (on distribution theory; Proceedings of the Second Canadian Mathematical Congress 1951). Dirac came to constrained Hamiltonian dynamics from a direction quite different from that of Rosenfeld or Bergmann. He was not interested in the effect of symmetries when passing from a Lagrangian to a Hamiltonian formulation of a field theory, be they general local symmetries (Rosenfeld) or general covariance in particular (Bergmann). Dirac’s focus was rather a subject that had been on his mind for quite some time: The relation between Hamiltonian dynamics and special relativity. Dirac viewed this as the central problem in constructing a (special) relativistic quantum (field) theory, a problem he still considered to be wide open, as he was not particularly inclined towards the renormalization methods being developed at the time.[8] He had already remarked in 1933:

Quantum mechanics was built up on a foundation of analogy with the Hamiltonian theory of classical mechanics. This is because the classical notion of canonical coordinates and momenta was found to be one with a very simple quantum analogue, as a result of which the whole of the classical Hamiltonian theory, which is just a structure built up on this notion, could be taken over in all its details into quantum mechanics. [...] [T]he Hamiltonian method is essentially non-relativistic in form, since it marks out a particular time variable as the canonical conjugate of the Hamiltonian function. (Dirac 1933b, 64.)

At the time, Dirac had believed that the way to proceed towards a relativistic quantum theory was consequently to find a new quantization procedure based on the classical Lagrangian, instead of the classical Hamiltonian, formalism. This was based on the realization that the action function appearing in the Lagrangian formalism, that is, the time integral of the Lagrangian itself, is a relativistic invariant. 16 years later, however, Dirac was considerably more optimistic about the reconcilability of Hamiltonian dynamics and relativity. The main reason for this appears to have been Dirac’s adoption of Weiss’s parameter formalism, which had been devised by Weiss precisely in order to obtain a covariant generalization of classical Hamiltonian dynamics, including canonical variables and Poisson brackets, as a starting point for the quantization of relativistic field theories.

In his Canadian lectures, Dirac went significantly beyond Weiss. Dirac emphasized that the parameter approach would turn the original space-time coordinates into dynamical variables in their own right. This came from a long tradition in Dirac’s thinking: Already in the mid-1920s (Dirac 1926), and then again in 1933 (Dirac 1933a), Dirac had explored the possibility of turning time from an evolutionary parameter into a dynamical variable in relativistic particle mechanics, an approach which is the particle-mechanical analog of the field-theoretical parameter formalism. As opposed to Bergmann, however, Dirac did not place any physical expectations in the newly dynamical nature of the coordinates: To him, this was merely a mathematical observation, one that ensured the (Lorentz) covariance of the scheme. In his 1949 lectures, Dirac further recognized that the use of the parameter formalism would imply the existence of constraints, including the Hamiltonian constraint, another insight going back to his early particle-mechanical exploration of the subject. This helped explain how the dilemma in the above quote was solved: The parameter formalism allowed a separation between (a) the Hamiltonian function giving the total energy or energy density (which now appeared as one canonical momentum among others, conjugate to the dynamical time coordinate) and (b) the generator of the equations of motion and the time evolution of physical quantities (which was now the task of the constraints, or, as Dirac had called them in 1933, the Hamiltonian equations).

It was on this basis that Dirac built up his approach to constrained dynamics. The structure he obtained was general enough to also accommodate constraints that did not arise from the use of parameters (and also to deal with a non-parameterized theory, involving only other constraints). Dirac mentioned this fact, even though he did not refer to any examples (back in 1933, however, he had mentioned the analogy with the constraints arising in electrodynamics), but Pirani and Schild, who were attending the conference, quickly picked up on the applicability of Dirac’s approach to General Relativity. Still, it should be emphasized that to Dirac the constraints arising from the parameters were the essential ones; additional constraints could simply be added on, as further identically fulfilled relations between canonical coordinates and momenta. Where these additional constraints might come from was of no concern to Dirac at the time and he did not discuss symmetry properties at all.[9] This of course made Dirac’s constrained dynamics look quite different from that of Bergmann, and one might spend a whole paper discussing the differences and similarities between the two. In the following, we will just briefly focus on the specific difference that was important for the work of Pirani and Schild, namely, that Dirac’s formalism provided an explicit construction principle for the Hamiltonian.

In working out the properties of their constraint g, Bergmann and Brunings had, quite naturally, considered it to be a function of canonical coordinates and momenta only, allowing for derivatives of the canonical coordinates with respect to the spatial parameters u, but not allowing for velocities, that is, derivatives of the canonical coordinates with respect to the time parameter t. However, the Hamiltonian density H is quite easily written down as a function of canonical coordinates, momenta, and velocities as

(31.1)

using Dirac’s notation, where the q_n are all the canonical coordinates (including the original space-time coordinates), the p_n are the conjugate momenta and the dot denotes differentiation with respect to the time parameter t. Now normally this is just an intermediate step in constructing the canonical Hamiltonian, the next step being the elimination of the velocities with the help of the defining equations for the momentum. This is, however, as we have already remarked in context of Rosenfeld’s work in Chapter 17, a non-trivial task. It was now Dirac’s innovation to consider Equation 31.1 not as an intermediate step—basically just a construction prescription—but rather as an actual expression for the Hamiltonian defined not on the usual phase space, but rather on a larger, 3 N-dimensional (with N the number of configuration variables) space, which also included the N velocities as independent variables. The usual Hamiltonian was then obtained by going to a 2 N-dimensional subspace, where all the defining equations for the momenta (including primary constraints) were fulfilled (this subspace of course corresponds to usual phase space if there are no constraints).

Dirac was now able to find another expression for the Hamiltonian density as a function of coordinates, momenta, and velocities:

(31.2)

where the v_m are some suitable functions, the important thing being that out of each summand one of the constraints \phi _m (the summation over m is over all constraints, those arising from parameterization and those intrinsic to the theory) can be factored out, the constraint being a function of coordinates and momenta alone.

Pirani and Schild, both in Toronto at the time, Schild working with Leopold Infeld and Pirani just having acquired his Master’s degree, attended Dirac’s Vancouver lectures in 1949. According to Pirani’s recollections,[10] Schild immediately realized the applicability of Dirac’s method to the quantization of general relativity.[11] Indeed, the paper of Bergmann and Brunings had appeared just a month before Dirac’s lecture, and it is quite probable that Schild had seen it and then in Vancouver realized how Dirac’s method might supply the Hamiltonian and the eighth constraint. He discussed the matter with Pirani, who promptly agreed to go with him to Pittsburgh (where Schild had just accepted a job at the Carnegie Institute of Technology) and become Schild’s first PhD student, the subject of his thesis being the quantization of general relativity. Pirani’s thesis (Pirani 1951) (which dealt, among other things, also with the inclusion of fermionic spinors) was not completed until 1951, but Pirani and Schild had already presented first results at the APS Meeting in New York (February 2–4, 1950, Schild and Pirani 1950) and submitted the paper reprinted here in Chapter 35 on February 14. By comparing the two forms of the Hamiltonian provided by Dirac, they were quite easily able to construct the first Hamiltonian for parameterized GR and thereby to also explicitly identify the eight primary constraints. Bergmann and his group were not far behind and published their Hamiltonian within the same year, constructed using a different procedure based on finding the “quasi-inverse” of the singular Legendre matrix (Bergmann et al. 1950).[12]

The success of Pirani and Schild and the Bergmann group in constructing a Hamiltonian for GR was initially seen as invalidating an approach based on the notion of a free graviton field, perturbed by its self-interaction and its interactions with other fields. Such an attempt at quantizing gravity had been worked out in the 1949 PhD thesis of Bryce DeWitt (né Seligman), a student of Schwinger’s, using the newly developed covariant techniques of his supervisor (DeWitt 1949). DeWitt wrote up a paper based on his thesis[13] in 1950 in which he duly acknowledged “the rigorous Pirani-Schild-Dirac scheme,” having been able to read a manuscript of the Pirani-Schild paper before publication. Still, DeWitt’s paper, which he submitted to the Physical Review, was essentially rejected by the referee, H. P. Robertson. In his referee report,[14] he stated

In view of the Pirani-Schild paper, which Seligman has seen—and remarks on p. 3, footnote, that this should eventually be carried out in terms of their more rigorous theory—it would seem to me better to suggest he carry out his work in terms of their theory.

DeWitt’s thesis work was never published, and he indeed turned to working on the Pirani-Schild formulation soon after (DeWitt 1952;DeWitt and DeWitt-Morette 1952).

However, as it turned out, the gravitational Hamiltonians of 1950 were not the last word on quantum gravity, with only details left to be filled out. The most immediate foundational reform was the abandonment of the parameter formalism by both groups within the same year. For Bergmann the parameter formalism had carried the hope of a quantum version of EIH, in which the space-time coordinates, which were dynamical variables in the parameter formalism, could be interpreted as the coordinates of a quantum mechanical point particle. That this will not work is rather obvious from a modern perspective: The parameterization introduces four additional constraints which reduce the number of dynamical degrees of freedom by exactly the number of dynamical variables one had formally gained by introducing the parameters in first place. But recall that Bergmann and Brunings had originally convinced themselves that there were no secondary constraints in their formalism; they were simply ignoring four of the constraints actually present in the formalism. And indeed, Bergmann’s abandonment of the parameter formalism appears to coincide with his realization that there were in fact additional secondary constraints.

In an interview, Joshua Goldberg placed the abandonment of the parameter formalism in the summer of 1950 or 1951,[15] with the published evidence clearly favoring the former (the first parameter-free paper by the Bergmann group was already submitted in May 1951, see Anderson and P. G. Bergmann (1951)). And in a short note to Schild, dated 16 November 1950,[16] Bergmann announced that his group had discovered additional constraints (i.e., had discovered the secondary constraints) and was working on a proof that there were as many additional constraints as there were primary ones. Such a proof was supplied a year later in the non-parameterized formalism (Anderson and Bergmann 1951), and really only makes sense therein, as there are more primary than secondary constraints in the parameter formalism. So there is strong circumstantial evidence that Bergmann’s abandonment of the parameter formalism was closely linked with the discovery of the secondary constraints, even though Bergmann never made that point explicitly in writing. In fact, he never even retracted his “no secondaries” argument in his published work. In the first detailed, published presentation of the parameter-free formalism (Penfield 1951), the new approach was merely presented as equivalent to the parameter formalism (for singularity-free gravitational fields), the manifest covariance of the latter thus ensuring the covariance of the former. And only another two years later, after the question of which were to be the actual physical observables in a constrained theory moved to the center of Bergmann’s attention, did he go on record stating that the parameterization did not, as he had hoped, introduce new observables (i.e., particle trajectories) into the theory (Bergmann and Schiller 1953).

The idea of introducing particle trajectories as new quantum operators via parameterization had never played a prominent role for Pirani and Schild. Parameterization had rather been a means (inherited from Dirac) for making the general covariance of the setup manifest, as it allowed for a definition of (parameter) time that did not presuppose any metric structure for space-time (which of course was to be a result of the field equations), but could be defined solely through the introduction of a family of three-dimensional hypersurfaces in the metric-free (Pirani used the term “amorphous” in his thesis) parameter space. When Schild received the above-mentioned note by Bergmann in November 1950, he had not yet considered abandoning parameters.[17] He and Pirani had (probably following Bergmann and Brunings) been convinced that there were no secondary constraints, without providing any argument for this statement. After receiving Bergmann’s note, Pirani attempted to calculate the secondary constraints in the parameterized formalism. This proved to be a formidable task, too formidable indeed: Pirani, in his thesis, merely presented some preliminary calculations in this direction, stating (p. 42) that “[t]he long and complicated expressions which make their appearances [...] led to a search for some way of simplifying the theory.” Pirani and Schild thus followed the lead of the Bergmann group, dropping the parameters, merely viewing them as a proof of the general covariance of the Hamiltonian formulation (Pirani 1951).[18]

Later work on the canonical quantization of gravity thus looked quite different from that done in these early works of 1949/50. In particular, as pointed out by Pirani to Schild in a letter from 4 December 1952,[19] the problem of integrating over singular regions in the action integral, which Bergmann had avoided by introducing parameters, remained. Indeed the parameter-free formalism strongly implied entirely doing away with the idea of particles as singularities (and thus the whole quantum EIH approach), and instead adopting more modern, (quantum) field-theoretical descriptions of matter. Still, this early work provided a basis for all later attempts at canonical quantization, introducing most of the concepts and terminology regarding the appearance of constraints.

References

Anderson, James L., and Peter G. Bergmann (1951). “Constraints in Covariant Field Theories.” Physical Review 83: 1018–25.

Bergmann, Peter G., Robert Penfield, Ralph Schiller, and Henry Zatzkis (1950). “The Hamiltonian of the General Theory of Relativity with Electromagnetic Field.” Physical Review 80: 81–88.

Bergmann, Peter G., and Ralph Schiller (1953). “Classical and Quantum Field Theories in the Lagrangian Formalism.” Physical Review 89: 4–16.

DeWitt, Bryce (1949). “I: The Theory of Gravitational Interactions. II: The Interaction of Gravitation with Light.” PhD thesis, Harvard.

——— (1952). “Point Transformations in Quantum Mechanics.” Physical Review 85: 653–61.

DeWitt, Bryce, and Cécile DeWitt-Morette (1952). “The Quantum Theory of Interacting Gravitational and Spinor Fields.” Physical Review 87: 116–22.

Dirac, Paul A. M. (1926). “Relativity Quantum Mechanics with an Application to Compton Scattering.” Proceedings of the Royal Society, Series A 111 (758): 405–23.

——— (1933a). “Homogeneous Variables in Classical Dynamics.” Mathematical Proceedings of the Cambridge Philosophical Society 29: 389–400.

——— (1933b). “The Lagrangian in Quantum Mechanics.” Physikalische Zeitschrift Der Sowjetunion 3 (1): 64–72.

Einstein, Albert (1916). “Die Grundlage Der Allgemeinen Relativitätstheorie.” Annalen Der Physik 49: 769–822.

Goldberg, Joshua (2005). “Syracuse: 1949–1952.” In Einstein Studies Volume 11: The Universe of General Relativity, edited by A. J. Kox and Jean Eisenstaedt, 357–71. Basel: Birkhäuser.

Halpern, Paul (2005). “Peter Bergmann: The Education of a Physicist.” Physics in Perspective 7: 390–403.

Kragh, Helge (1990). Dirac: A Scientific Biography. Cambridge: Cambridge University Press.

Penfield, Robert (1951). “Hamiltonians Without Parameterization.” Physical Review 84: 737–43.

Pirani, Felix A. E. (1951). “On the Quantization of the Gravitational Field of General Relativity.” PhD thesis, Carnegie Institute of Technology.

Pitts, J. Brian (2014). “Change in Hamiltonian General Relativity from the Lack of a Time-Like Killing Vector Field.” Studies in History and Philosophy of Modern Physics 47: 68–89.

Proceedings of the Second Canadian Mathematical Congress (1951). Toronto: University of Toronto Press.

Rao, C. Radhakrishna, and Sujit Kumar Mitra (1971). Generalized Inverse of Matrices and Its Applications. New York: Wiley.

Salisbury, Donald (2006). “Peter Bergmann and the Invention of Constrained Hamiltonian Dynamics.” arXiv:Physics/0608067v1.

Salisbury, Donald, and Kurt Sundermeyer (2017). “Translation into English of Léon Rosenfeld’s "Zur Quantelung Der Wellenfelder" with Commentary.” European Physical Journal H 42 (1): 23–61.

Schild, Alfred, and Felix A. E. Pirani (1950). “On the Quantization of Einstein’s Gravitational Field Equations.” Physical Review 78: 329.

Schweber, Silvan S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton: Princeton University Press.

Williamson, Ralph E. (1951). “Fred Hoyle’s Universe.” The Journal of the Royal Astronomical Society of Canada 45: 185–89.

Footnotes

[1] See Schweber (1994).

[2] On Bergmann’s background, see Halpern (2005).

[3] Indeed, Bergmann’s paper contains the cryptic remark that general relativity had “so far not successfully absorbed the existence of quantities possessing half-odd spin.” It is an open historical problem what this statement refers to.

[4] A Lagrangian for General Relativity that contained only first-order derivatives of the metric had first been introduced by Einstein (1916), the so-called \Gamma -\Gamma-Lagrangian, given by Einstein originally only for a specific choice of coordinates. Rosenfeld had actually also considered Lagrangians that change by a total divergence, but only for “internal” symmetries that do not act on the space-time coordinates. It was thus not directly applicable to Lagrangians that changed by a total divergence under general coordinate transformations, but it did apply to Rosenfeld’s tetrad Lagrangian, that changed by a total divergence only under local Lorentz transformations. Rosenfeld’s tetrad Lagrangian was a scalar density under general coordinate transformations. His method can be straightforwardly extended to the case of coordinate transformations, as shown in Salisbury and Sundermeyer (2017).

[5] Concerning these functionals, which were to represent physical quantities, Bergmann and Brunings made the crucial observation in this paper that not all functions of the original field variables and their first derivatives (velocities) possessed a counterpart in phase space. They identified the necessary and sufficient condition for converting a function in configuration-velocity-space into a phase-space function (such functions need to be, in modern parlance, constant along all of the null directions of the singular Legendre matrix) and dubbed quantities that satisfied this requirement “dynamical variables”.

[6] This fact has already been pointed out by one of us (Salisbury 2006). It is hard to say precisely how Bergmann and Brunings came to this erroneous conclusion, since they do not give the details and there is no equation in the paper that is actually wrong. The erroneous statement is that “[t]he proof follows immediately if the functional \mathcal {G} [the linear combination of primary constraints] is substituted in the expression for the Poisson bracket (3.23) [...]” A simple counterexample is to be found in the toy model employed in Pitts (2014), which can easily be parameterized. We would like to thank James Brian Pitts for suggesting this check upon Bergmann and Brunings’s claim.

[7] That the Hamiltonian of GR would be a constraint even in the unparameterized formulation, the so-called “problem of time,” was only gradually realized in the course of the 1950s.

[8] See, e.g., Kragh (1990, 183).

[9] Joshua Goldberg (2005) has thus labelled Dirac’s approach as algebraic (concerned with the constraints themselves), as opposed to the group theoretical (concerned with the underlying symmetries) approach of Bergmann (and also Rosenfeld).

[10] Interview by DR, 23 June 2011, https://www.aip.org/history-programs/niels-bohr-library/oral-histories/34463 (accessed 21 July 2017).

[11] Dirac had apparently not realized this and Schild brought it to his attention at the time. His wife, Winnie Schild, told one of the authors (DS) of her husband’s pride in having achieved Dirac’s enduring respect through this observation. More importantly, the fact that Dirac had not made the connection underlines that it was not as obvious as Pirani recalled it to be in the interview with DR, making Schild’s previous exposure to Bergmann and Brunings’s work all the more probable.

[12] For more on the mathematical theory (and a brief history) of the generalized inverses of singular matrices, see Rao and Mitra (1971). In the nomenclature adopted there, the Bergmann group was working with a (not uniquely defined) reflexive g-inverse of the Legendre matrix.

[13] A manuscript is in the possession of his widow, Cecile Morette-DeWitt.

[14] 17 May 1950, H. P. Robertson Papers, Caltech

[15] Interview by DR and DS, 21 March 2011, https://www.aip.org/history-programs/niels-bohr-library/oral-histories/34461 (accessed 21 July 2017).

[16] Alfred Schild Papers, Dolph Briscoe Center for American History, The University of Texas at Austin.

[17] In the Schild Papers, there is a letter from Pirani dated 22 January 1951, in which the use of the parameter-free formalism is already discussed as a matter of course, seemingly contradicting the above statement. However, in this letter Pirani makes reference to a radio broadcast by Ralph Williamson, which was only broadcast in June 1951 (Williamson 1951), strongly implying a redating of this letter to 1952, making this an instance of the classic mistake of still dating with the old year in January.

[18] The resultant simplifications also allowed Pirani and Schild to identify a mistake in their first paper (see the January 1951/52 letter from Pirani to Schild mentioned in footnote 17): Equation 62 is wrong, no matter where one chooses to close the square bracket: If it is closed at the end of the expression, the sign of the last summand should be changed from plus to minus. If it is closed before the last summand, then that summand needs an additional factor of l_{\gamma } l^{-2}.

[19] Alfred Schild Papers

32. Peter G. Bergmann (1949): Non-Linear Field Theories

Peter G. Bergmann (1949). Non-Linear Field Theories. Physical Review, 75: 680–685.

33. Peter G. Bergmann and Johanna H. M. Brunings (1949): Non-Linear Field Theories II: Canonical Equations and Quantization

Peter G. Bergmann and Johanna H. M. Brunings (1949). Non-Linear Field Theories II: Canonical Equations and Quantization. Reviews of Modern Physics, 21: 480–487.

34. P. A. M. Dirac (1950): Generalized Hamiltonian Dynamics

P. A. M. Dirac (1950). Generalized Hamiltonian Dynamics. Canadian Journal of Mathematics, 2: 129–148.

35. Felix Pirani and Alfred Schild (1950): On the Quantization of the Gravitational Field Equations

Felix Pirani and Alfred Schild (1950). On the Quantization of the Gravitational Field Equations. Physical Review, 79: 986–991.