A roadmap for Feynman's adventures in the land of gravitation
aa r X i v : . [ phy s i c s . h i s t - ph ] F e b A roadmap for Feynman’s adventures in the land ofgravitation
Marco Di Mauro , Salvatore Esposito , Adele Naddeo Dipartimento di Fisica “E.R. Caianiello”, Universit`a di Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy. INFN Sezione di Napoli, Via Cinthia, 80126 Naples, Italy.
February 23, 2021
Abstract
Feynman’s multifaceted contributions to gravitation, as can be inferred from several published andunpublished sources, are here reviewed. Feynman thought about this subject at least from late 1954to the late ’60s, giving several pivotal contributions to it, and as a byproduct, also to Yang-Millstheory. Even though he only published three papers on gravity, much more material is available,especially the records of his many interventions at the Chapel Hill conference in 1957, which arehere analyzed in detail. In that conference, Feynman showed that he had already developed muchof his picture of gravity, and in addition he expressed deep thoughts about fundamental issues inquantum mechanics, such as superpositions of the wave functions of macroscopic objects and therole of the observer, which were suggested by the problem of quantum gravity. Moreover, Feynmanlectured on gravity several times. Besides the famous lectures given at Caltech in 1962-63, heextensively discussed this subject in a series of lectures given at the Hughes Aircraft Company in1966-67, whose focus was astronomy and astrophysics. Here Feynman gave a somewhat simplifiedexposition with respect to the Caltech lectures, but with many original points. All this materialallows to reconstruct a detailed picture of Feynman’s ideas on gravity and of its evolution until thelate sixties. The main points are that gravity, like electromagnetism, has a quantum foundation,so that classical general relativity has to be regarded as emerging from an underlying quantumtheory in the classical limit, and that this quantum theory has to be investigated by computingphysical processes, as if they were easily measurable. The same attitude is shown with respect togravitational waves, as evident both from the Chapel Hill records and an unpublished letter writtento V. Weisskopf. As a bonus, an original approach to gravity, which closely mimics (and probablywas inspired by) the derivation of the Maxwell equations he gave in that period, is hinted at in theunpublished lectures.
Feynman’s vision of physics was not sectorial, but rather based on the belief that it constitutes a singleunified corpus, reflecting the unity of nature. In particular, he was one of the first to concern about1he relation of gravitation to the rest of physics, in a period (the early 1950s) where general relativitypractitioners still tended to be isolated from mainstream research, while the so-called renaissance of general relativity was still some years in the future [3]. In particular, starting at least from late1954 (as recalled in [4, 5]), he started thinking deeply about gravitation, giving several fundamentalcontribution to it in the following years, until the late sixties, where he apparently lost interest in thesubject. The sticky bead argument [1], the Feynman rules for general relativity and the associatedghosts [2], the Caltech Lectures on Gravitation [6], and the Feynman-Chandrasekhar instability ofsupermassive stars [7], are now part of the common lore about classical and quantum gravity.In this paper we retrace in some detail the full development of Feynman’s thoughts on this subject.As a starting point, we choose the 1957 Chapel Hill conference, where Feynman’s thoughts on gravitywere publicly expressed for the first time, and the first written records of them were put down. Inthat conference, which was pivotal in triggering the renaissance of general relativity, the gravitationalcommunity delineated the tracks along which subsequent work would develop. Besides cosmology,which at the time was still considered “a field on its own, at least at present, not intimately connectedwith the other aspects of general relativity” ([8], p. 352), the main headings were the following three[8]: classical gravity, quantum gravity, and the classical and quantum theory of measurement (as alink between the previous two topics). The records of Feynman’s interventions at that conference,in fact, show that he had already deeply thought, and performed computations, about all threetopics, focusing on classical gravitational waves, on the arguments in favor of quantum gravity fromfundamental quantum mechanics, and finally on quantum gravity itself. We shall therefore develop ourtreatment along these lines, starting with Feynman’s interventions at Chapel Hill and then followingthe development of his thoughts in the subsequent years. We include in particular some previouslyunavailable unpublished material. This material refers to two sets of lectures, which Feynman gavein the 60’s at the Hughes Aircraft Company, which have recently been made available on the web .In particular, in the 1966-67 set of lectures [10], which were devoted to astronomy, astrophysics andcosmology (adopting an unusual ordering of topics, roughly by inverse scale, starting from cosmology,then going through stars, their evolution, active galactic nuclei, and ending with the solar system),there is a fairly dense treatment of general relativity, with the purpose of applying it to cosmologyand to astrophysics. Besides many similarities with the more famous 1962-63 Caltech lectures [6], thistreatment displays several original points, probably due to Feynman’s thoughts in the years betweenthe two courses. In those years, as we reported elsewhere [11, 12], another of Feynman’s concernswas the development of a different derivation of Maxwell’s equations, thus finding an original wayof addressing the teaching of electromagnetism. Probably inspired by these thoughts, in the Hugheslectures Feynman suggested that a similar approach, with suitable modifications, could be adopted “The problem of the relation of gravitation to the rest of physics is one of the outstanding theoretical problems ofour age”([1], p. 15); “My subject is the quantum theory of gravitation. My interest in it is primarily in the relation ofone part of nature to another” ([2], p. 697) Actually, since Gell-Mann wrote that Feynman had “made considerable progress”, it is very likely that he had alreadybeen working on gravity for some years in 1954, probably starting shortly after taming quantum electrodynamics, in theearly 50s. The story of how Feynman got involved in teaching at the Hughes Aircraft Company is briefly told in [9]
The Chapel Hill Conference in 1957 [1] was the first important international conference on generalrelativity (and was called GR1 henceforth), where the new generation of physicists, including theDeWitts, Feynman, Wheeler, Deser, Misner, Pirani, etc. participated, and it contributed to definingthe trends for most of the subsequent research in classical and quantum gravity, and to the recognitionfor general relativity to be considered as part of physics. Such a recognition, whose need was widely feltamong the practitioners, was indeed the main aim of the conference, as its title ( The Role of Gravitationin Physics ) clearly shows. An excellent account of the events that brought to the foundation of theInstitute of Field Physics and to the conference can be found in the introduction of [1], written forthe occasion of its recent re-publication.
During the conference, Feynman proposed the well known sticky bead argument to simply and in-tuitively argue that gravitational waves carry energy, and therefore are a real physical phenomenon,not just coordinate artifacts. Also, he characterized Everett’s interpretation of quantum mechanics, For the full list see [1], pp. 39-40. Actually, the first conference devoted entirely to relativity was the Berne Jubilee conference of 1955 (sometimescalled GR0) [14], which however mostly involved physicists of the older generation, excluding in particular most of theyounger American recruits.
The general covariance of the theory of general relativity makes it difficult – in the absence of aninvariant characterization – to distinguish between real physical phenomena and artifacts due to thechoice of the coordinate system, which would disappear by changing the latter. At the time of theChapel Hill conference, this was just the case for gravitational waves, which had been studied until4hen only in special coordinate systems . In 1916 [16] and 1918 [17], Einstein first found that hisgravitational field equations, when linearized, admit wave-like solutions traveling with the speed oflight, by adopting a particular coordinate system, and in addition showed that at least some of thesolutions he found did carry energy, and he gave a formula for the leading order energy emission rate,the famous quadrupole formula . However, he encountered some difficulties in the definition of suchenergy, due to the fact, already known to him, that energy is not localizable in general relativity. In1922, the reality of gravitational waves was doubted by A. S. EddingtonThe potentials g µν pertain not only to the gravitational influence which has objectivereality, but also to the coordinate-system which we select arbitrarily. We can “propagate”coordinate-changes with the speed of thought , and these may be mixed up at will withthe more dilatory propagation discussed above. There does not seem to be any way ofdistinguishing a physical and a conventional part in the changes of the g µν (cf. [18], p.130, emphasis in the original) .Further doubts concerned the possibility that gravitational waves were just artifacts of the linearizedapproximation, which would disappear once the full theory was considered. In fact, Einstein himself[20], while attempting, along with Nathan Rosen, to find plane wave solutions of the full nonlinearfield equations of gravitation, believed to have shown that the full theory predicted no gravitationalwaves . However, after a mistake was found, Einstein and Rosen finally discovered a rigorous solutiondescribing waves radiated off the axis of a cylinder [22]. This did not settle the issue yet, however,since the problem of the physical effects of gravitational waves remained unsolved, as well as the issueof whether or not such gravitational waves carry energy. Such problems arose as a consequence of thelack (at that time) of a general relativistic theory of measurement, along with the already mentioneddifficulties inherent to the definition of energy in general relativity. In particular, the wave solutionsfound by Einstein and Rosen seemed not to carry energy, as still argued by Rosen himself in 1955([14], pp. 171-174), and it was not known whether this came from their unbounded nature; also, itwas as well not known whether spherical wave solutions do exist, describing energy carried away by alocalized center of radiation.The dilemma thus persisted that gravitational waves could exist only as mathematical objects, withno actual physical content, and all the above problems remained unsolved until the appearance of F.Pirani’s work in 1956 [23, 24] and the subsequent discussions which took place at Chapel Hill in 1957[1], where this work was presented. While investigating the physical meaning of the Riemann tensorin terms of the geodesic deviation equation, indeed, Pirani managed to give an invariant definition of For an excellent account of the fascinating history of gravitational waves, which adds considerable detail to whatfollows, see [15]. Actually, Eddington was not questioning the existence of gravitational disturbances traveling with a finite speed,which was required by special relativity, but rather Einstein’s procedure, in which the speed of light seemed to be putin by hand by the particular coordinate choice he adopted. In fact, in a subsequent paper [19], Eddington showedmore rigorously that some of Einstein’s wave solutions were not spurious, i.e. were not just flat space in curvilinearcoordinates, and did propagate with the speed of light. In the same paper, he corrected a minor error in Einstein’squadrupole formula, and applied it to compute the radiation reaction for a rotator. See also [21], letter 71. As wewill see, this was just Feynman’s starting point in proposing his famous sticky bead argument .An additional unsolved issue, related to the above, came to physicists’ attention, namely thecomputation (in some approximation) of the energy loss rate in a binary star system due to emissionof gravitational waves, from which a further question emerged, concerning whether such waves carryenergy proportional to the square of their amplitude. The problem was that such a bound systemcannot be described in the linearized approximation. Such a computation was moreover complicatedby the fact that the radiation reaction on the stars had to be taken into account, hence the orbitsare expected to shrink. It was not known whether Einstein’s quadrupole formula correctly describedthe energy loss beyond the linear approximation. Indeed it is very difficult to analytically solve theEinstein equations in the case of a strongly gravitating source, in order to compute the amplitudeof gravitational waves. The best approximation scheme, suitable to gravitationally-bound systems(which represent the majority of known astronomical sources), is the post-Newtonian one (for reviewssee Refs.[26, 27]). It assumes, as small parameters, the magnitude of the metric deviation from thebackground Minkowski metric and the squared ratio (cid:0) uc (cid:1) between the typical velocity of the system u and the speed of light c . For instance, in the case of a binary system the typical velocity is theorbital velocity while the Newtonian potential plays the role of the deviation form the flat metric.The result is a Newtonian description in the lowest order while general relativistic effects arise ashigher order perturbations. In such an approximation the radiative effects emerge from the order (cid:0) uc (cid:1) , i.e. the 2 . . And of course, there were the ubiquitous ambiguities indefining energy in general relativity. The problem was first addressed by Landau and Lifshitz in thefirst edition of their textbook [28], where the quadrupole formula was applied. But it was not clearwhether their calculation was correct. In fact, at the time of the Chapel Hill conference, there stillwas no consensus about the result of such a computation, and not even on the fact that gravitationalwaves were radiated at all. And, differently from the first issue, this one was not settled at ChapelHill, but continued to be controversial for many years, because of several conceptual issues [15]. It wasindeed addressed several more times, with different approaches. Interestingly, one of the most commonobjections towards the existence of gravitational radiation reaction was the lack of any evidence thatadvanced potentials did not play any role in the theory. Some people believed that maybe gravityworked like a Wheeler-Feynman absorber theory, in which the relevant solution was a combination ofretarded plus advanced potential (despite the nonlinearity of the theory prevented a combination oftwo solutions to be a rigorous solution), but unlike the case of electromagnetism, the weakness of theforce prevented the existence of absorbers capable of breaking time symmetry, as in electromagnetism.Apparently, despite having developed the absorber theory for electromagnetism, Feynman did notbelieve in such ideas, since, as we shall see, he solved the wave equation in linearized gravity in terms This same reasoning was also applied to the cylindrical waves of Einstein and Rosen by J.Weber and J.A. Wheelerin the special issue of Reviews of Modern Physics devoted to the conference [25]. This is the lowest nontrivial order with an odd power of v , which signals the dissipation of energy due to radiation.
6f retarded potentials only. Also, he did not share the concerns towards this issue, which he consideredas solved after he derived the quadrupole formula and used it to compute the energy loss rate.All these hot topics captured the interest at the Chapel Hill Conference, as clearly recalled byBergmann in his Summary [8]:In view of our interest in the role that gravitation, and particularly its quantum properties,may play in microphysics, the existence and the properties of gravitational waves representan issue of preeminent physical significance.And at that conference, Feynman contributed to both, as we shall now see.
The relevance of the actual existence of gravitational waves emerged properly from the discussionfollowing Feynman’s thought experiment which we discuss in Sec. 3. Feynman arrived one day laterat the conference and hence did not attend the session dedicated to gravitational waves. Apart fromthe obvious meaning that the presumed existence could have for the classical theory, the questionof whether the gravitational field had to be quantized or not was just starting to be considered.Although trying to argue for the need of quantizing the gravitational field also without the presence ofgravitational waves, Feynman did come out with a simple physical argument in favor of their existence,i.e. the celebrated sticky bead argument ([1], p. 260 and pp. 279-281).Feynman argued that gravitational waves, if they exist, must carry energy and, along with theirexistence as solutions of the gravitational equations (albeit in the linear approximation), this wasenough for him to be confident in the possibility of their actual generation: “My instincts are that ifyou can feel it, you can make it” ([1], p. 260). Feynman’s reasoning went as follows:Suppose we have a transverse-transverse wave generated by impinging on two masses closetogether. Let one mass A carry a stick which runs past touching the other B . I think Ican show that the second in accelerating up and down will rub the stick, and therefore byfriction make heat. I use coordinates physically natural to A , that is so at A there is flatspace and no field [...]. ([1], p. 279).Then he recalled the previous result by Pirani [23, 24] (which was also presented earlier at the confer-ence, cf. [1], p. 141), stating that the displacement η of the mass B (measured from the origin A of thecoordinate system) in the path of a gravitational wave satisfies the following differential equation: ¨ η a + R a b η b = 0 , ( a, b = 1 , ,
3) (1) R being the curvature tensor at A . The curvature does not vanish for the transverse-transverse gravitywave but oscillates as the wave goes by, so that Eq. (1) predicts that the particle vibrates a littleup and down, hence rubs the stick and finally produces heat by friction. This means that the stick This follows from the geodesic deviation equation ([1], p. 141; [23, 24]). π √ mMm + M (cid:16) uc (cid:17) . (2)Here m and M are the masses of the two bodies, while u is the magnitude of their relative velocity.For the earth-sun system, this formula produces just a tiny effect leading to a huge order of magnitude(about 10 years) for the lifetime of the motion of the earth around the sun. Indeed, in whole analogywith the radiation arising from an accelerated charge in a Coulomb field, gravitational radiationis expected to be emitted by the earth-sun binary system due to mass acceleration under mutualgravitation, thus making earth to slowly spiral into the sun after 10 years.The above argument was actually based on a detailed analysis performed by Feynman some timebefore ([1], p. 280), but the whole calculation (with some differences, according to Feynman) wouldhave been reported only four year later in a letter written to V. Weisskopf [29] (see below). Apart of that calculation can also be found in the 16-th of his Caltech Lectures [6]. Here Feynman’sreasoning develops in analogy with electrodynamics, but taking into account the fact that in the caseof gravitation the source is a tensor S µν rather than a vector A µ , so that the following differentialequation has to be solved to find classical gravitational waves: (cid:3) ¯ h µν = λS µν . (3)where as usual ¯ h µν represents the metric perturbation (trace-reversed, cf. Eq. (9) below) and thede Donder gauge ∂ µ ¯ h µν = 0 has been adopted. In the case in which all quantities are periodic, withfrequency ω , the solution at a point 1, which is located at a distance much greater than the dimensionsof the region where S µν is expected to be large, is:¯ h µν ( ~r ) = − λ πr e iωr Z d ~r S µν ( ~r ) e − i ~K · ~r , (4)where | ~r | ≫ | ~r | ; thus the first non vanishing term in the sequence of integrals corresponding to anexpansion of the exponential is a quadrupolar one. This approximation holds on for nearly all casesof astronomical interests, where wavelengths are much longer than system’s dimensions, such as forinstance double stars or the earth-sun system. Feynman’s attention then concentrates on the issue ofhow to compute the power radiated by the above waves, whose result for a circularly rotating doublestar should coincide with formula (2).H. Bondi and J. Weber were among the people who attended the Chapel Hill conference. Bondilikely envisaged independently an argument similar to that of Feynman, and shortly after the con- This is proved by several of his remarks at the conference, such as the following one, made after Pirani’s talk ([1], detection of gravitational waves.
Although Feynman never published anything on the issue, a complete and more systematic descriptionof his Chapel Hill proposal can be found in the already mentioned letter he wrote to Victor Weisskopfin February 1961 [29], and subsequently included in the material distributed to the students attendinghis Caltech lectures [6] in 1962-63 .It was this entire argument used in reverse that I made at a conference in North Carolinaseveral years ago to convince people that gravity waves must carry energy. [29].In the letter, Feynman provided a much more accurate description of the gravitational wave detectorhe described at Chapel Hill, also deriving the formula for the power radiated by a gravitational wavein the quadrupole approximation. It reads, for a periodic motion of frequency ω [29]:15 Gω X ij (cid:12)(cid:12)(cid:12) Q ′ ij (cid:12)(cid:12)(cid:12) , (5)where Q ′ ij = Q ij − δ ij Q kk , Q ij = P a m a R ai R aj being the quadrupole moment of mass and the RMSaverage is taken. In particular, for a circularly rotating double star, formula (2) is recovered. Thus,Feynman’s treatment applied the quadrupole formula to a binary system, in this being analogous tothe Landau and Lifshitz one. Feynman’s treatment is anyway simpler, since he manages by a clevertrick to avoid referring to the energy-momentum pseudo-tensor (this is the “simpler argument” wereferred to above). Apparently, at Chapel Hill, he derived the expression for the radiated energy fromthe detailed study of the detector, while in the letter the opposite route is taken (hence he says thatthe argument is “used in reverse”). The question of course arises, whether he did take inspiration by p.142): “Can one construct in this way an absorber for gravitational energy by inserting a dη/dτ term, to learn whatpart of the Riemann tensor would be the energy producing one, because it is that part that we want to isolate to studygravitational waves?”. The absorber of gravitational energy is, of course, nothing but the “stickiness” of Feynman’ssticky bead. In fact, the computations in the letter must have been somewhat simpler, since Feynman admits that “Only as Iwas writing this letter to you did I find this simpler argument”.
One of the most debated questions at the Chapel Hill conference was whether the gravitational fieldhad to be quantized at all. This was seen as one of the crucial matters in view of the merging ofgeneral relativity with the rest of physics, especially with the then thriving field of elementary particlephysics, which of course was dominated by quantum mechanics. As remarked by Bergmann ([1], p.165): Physical nature is an organic whole, and various parts of physical theory must not beexpected to endure in “peaceful coexistence.” An attempt should be made to force separatebranches of theory together to see if they can be made to merge, and if they cannot beunited, to try to understand why they clash. Furthermore, a study should be made of theextent to which arguments based on the uncertainty principle force one to the conclusionthat the gravitational field must be subject to quantum laws: (a) Can quantized elementary10articles serve as sources for a classical field? (b) If the metric is unquantized, would thisnot in principle allow a precise determination of both the positions and velocities of theSchwarzschild singularities of these particles?Besides the many technical talks and comments, where the main approaches to quantum gravity(namely the canonical, the functional integral and the covariant perturbative approaches), pursuedin the following decades, were outlined, a lot of efforts were devoted to fundamental and conceptualquestions, also in view of the feeling that physicists shouldattempt to keep physical concepts as much as possible in the foreground in a subjectwhich can otherwise be quickly flooded by masses of detail and which suffers from lack ofexperimental guideposts ([1], p. 167).Hence the problem of quantum measurement was discussed at long during the conference. As sum-marized by Bergmann, the main conceptual question was:What are the limitations imposed by the quantum theory on the measurements of space-time distances and curvature ([1], p. 167)?or, equivalentlyWhat are the quantum limitations imposed on the measurement of the gravitational massof a material body, and, in particular, can the principle of equivalence be extended toelementary particles ([1], p. 167)?As pointed out by the editors of Ref. [1], the answer couldn’t be simply given in terms of dimensionalarguments, in the sense that the Planck mass does not constitute a lower limit to the mass of a particlewhose gravitational field can in principle be measured. A simple argument ([1], pp. 167-8) shows thatthe gravitational field of any mass can in principle be measured, thanks to the “long tail” of theNewtonian force law.
Feynman’s views were very close to those of Bergmann concerning such general matters (while theydisagreed on more technical ones). Indeed, as evident from his interventions in the many discussionsessions devoted to this question, Feynman was convinced that Nature can not be half classical andhalf quantum, so that he opposed those proposing that maybe gravity should not be quantized, aposition obviously conflicting with his unified vision of Nature. Feynman’s arguments in favor ofthe quantization of the gravitational field were quite different from the ones that became popularmore later among most practitioners, involving the assumption that at the Planck scale gravity hasto dominate over all other interactions. Feynman thought that it was not necessary to go to suchridiculously high and unfathomable scales to see quantum effects involving gravity. More in detail, heproposed a thought experiment showing that, by assuming that quantum mechanics holds for objectsbig enough to produce a gravitational field (an assumption in which he definitely believed, since the11pposite would have required a modification of quantum mechanics [41]), then the only way to avoidcontradictions is that the gravitational field has to be quantized ([1], pp. 250-252). Alongside withthe consideration that a mass point giving rise to a classical Schwarzschild field very unlikely wouldobey Heisenberg’s uncertainty relations, a more pragmatic motivation was the hope (also expressedby people like Pauli) that quantization of the metric would help resolving the divergencies present inquantum field theory and, in this way, it would have a key relevance also for the theory of elementaryparticles. This view emerges, in particular, from Feynman’s criticism to Deser’s different proposal,where all other fields should be quantized in a given geometry, the quantization of the metric fielditself being the final step of the whole procedure [8].If one started to compute the mass correction to, say, an electron, one has two propagatorsmultiplying one another each of which goes as 1 /s and are singular for any value of the g ’s.Therefore, do the spatial integrations first for a fixed value of the g ’s and the propagatoris singular, giving δm = ∞ . Then the superposition of various values of the g ’s is stillinfinite ([1], p. 270).Feynman’s approach to quantum gravity, as we shall see, was in fact fully quantum from the beginning. The discussions about the need to quantize gravity, along with the related issue of whether objects,that are large enough to generate measurable gravitational fields, are subject to quantum mechanics,were one of the few places where Feynman was directly involved in the interpretation of quantummechanics [41] or, at least, where his thoughts about the subject were externalized.Based essentially on thought experiments, his arguments stimulated a wide debate on the measure-ment problem in quantum mechanics and on the existence and the meaning of macroscopic quantumsuperpositions [1]. The debate further grew up in Session VIII of the Chapel Hill Conference, wherethe discussion focused on the contradictions eventually arising in the logical structure of quantumtheory without assuming gravity quantization. Indeed, as pointed out by DeWitt ([1], p. 244), thechoice for the expectation value of the stress-energy tensor as the source of the gravitational field,when quantized matter fields are present, may lead to difficulties, since measurements made on thesystem may change this expectation value and hence the gravitational field.The validity of the classical theory of gravitation relies on the smallness of the experimentalfluctuations at the scale where gravitational effects become sizeable. Here Feynman came into playwith his first argument, concerning a two-slit diffraction experiment with a mass indicator behind thetwo-slit wall, i.e. a gravitational two-slit experiment. If one is working in a space-time region whosedimensions are of order L in space and L/c in time, the uncertainty on the gravitational potentialis in general ∆ g = q hGc L = L P L (as argued for example by Wheeler in [1], pp. 179-180, on the basisof an argument involving a path integral for the gravitational field). The potential generated by amass M within the volume L is g = MGLc , hence in this case ∆ g = ∆ MGLc , from which by comparisonwith the general expression one could infer a mass uncertainty ∆ M = c L P G ≈ − grams, if the time12f observations is less than L/c . On the other side, by allowing an infinite time, one would obtain M with infinite accuracy. Feynman deduced that the last option couldn’t take place for a mass M fed into the above two-slit apparatus, so that the apparatus will not be able to uncover this difficultyunless M is at least of order 10 − grams. His conclusion was that:Either gravity must be quantized because a logical difficulty would arise if one did theexperiment with a mass of order 10 − grams, or else that quantum mechanics fails withmasses as big as 10 − grams ([1], p. 245).The discussion then briefly dealt with some formal matter and with the meaning of the equivalenceprinciple in quantum gravity, with Salecker proposing a thought experiment which apparently lead toa violation of the equivalence principle in the quantum realm. Finally, due to a remark by Saleckerhimself, the discussion again focused on the topical question, i.e. why gravitational field needs to bequantized at all. Indeed, Salecker pointed out how charged quantized particles may act as a source of anunquantized Coulomb field within an action-at-a-distance picture, perhaps hinting (as commented byan Editor’s Note (see [1], p. 249)) to a similar situation in the case of a gravitational field. Subsequentconsiderations by Belinfante suggested the quantization of the static part of the gravitational field aswell as of the transverse part (which describes the gravitational radiation), in order to avoid difficultiesarising from the choice of an expectation value (i.e. that of the stress-energy tensor) as the source ofthe gravitational field, in full agreement with the previous argument by DeWitt. Here, as noted byZeh [41], Belinfante’s understanding of the wave function as an epistemic concept clearly emerges, andFeynman replies with his second thought experiment, a Stern-Gerlach experiment with a gravitationalapparatus, which reveals his completely different ideas on the meaning of quantization and on the roleof wave function.Feynman’s argument deals with a spin-1 / . However, thanks to its macroscopic size, the ball willbe able to produce a gravitational field and such a field may be useful in order to move a probe ball,as well as to use the connections with this probe as a measuring apparatus. Thus the gravitationalfield acts as a channel between the object and the observer. This reasoning leads Feynman to thefollowing conclusion about gravity quantization:Therefore, there must be an amplitude for the gravitational field, provided that the am-plification necessary to reach a mass which can produce a gravitational field big enough toserve as a link in the chain does not destroy the possibility of keeping quantum mechanics As pointed out by Zeh [41], Feynman’s description of the measurement process is very close to the standard mea-surement and registration device proposal by von Neumann [42].
Feynman then focuses on the problem of wave function collapse and hints to decoherence as a possiblesolution:There is an amplitude that it is up and an amplitude that it is down – a complex amplitude– and as long as it is still possible to put those amplitudes together for interference youhave to keep quantum mechanics in the picture ([1], p. 252).Feynman argues that the wave packet reduction acts somewhere in his experimental apparatus thanksto the amplifying mechanism, so that amplitudes become probabilities in the presence of a hugeamount of amplification (via the gravitational field of the ball). Then he proceeds by wondering if itmight be possible to design an experiment able to invert the above amplification process. Subsequentcritical remarks by Rosenfeld and Bondi lead Feynman to envisage a kind of quantum interferencein his experiment, by allowing gravitational interaction between macroscopic balls to be described bymeans of a quantum field with suitable amplitudes taking a value or another value, or to propagatehere and there. Clearly, as suggested by Bondi, one has to remove any irreversible element such as, forinstance, the possibility that gravitational links radiate. This is probably another hint to the possiblerole of dissipation or decoherence in destroying quantum interference, but Feynman and Bondi hereonly speak about classical irreversibility [41], the concept of decoherence as source of smearing offphase relations, as well as the transition to the classical picture and its environmental origin, beingstill unclear (for a historical and research account see Ref. [43] and references therein). By arguingagain on gravity quantization in relation to wave function collapse, Feynman claims that (emphasisin the original):There would be a new principle ! It would be fundamental ! The principle would be: – roughly : Any piece of equipment able to amplify by such and such a factor (10 − grams orwhatever it is) necessarily must be of such a nature that it is irreversible . It might be true!But at least it would be fundamental because it would be a new principle. There are twopossibilities. Either this principle – this missing principle – is right, or you can amplify toany level and still maintain interference, in which case it’s absolutely imperative that thegravitational field be quantized... I believe ! or there’s another possibility which I have’tthought of ([1], pp. 254-255).The discussion on possible sources of decoherence goes on further with Feynman assuming that quan-tum interference might eventually take place with a mass of macroscopic size, i.e. about 10 − gram or14 gram, and hinting to the possible role of gravity in destroying quantum superpositions. The sameargument is later developed by Feynman in his Caltech lectures on gravitation [6], where he dealswith philosophical problems in quantizing macroscopic objects and comments about a possible gravityinduced failure of quantum mechanics:I would like to suggest that it is possible that quantum mechanics fails at large distancesand for large objects. Now, mind you, I do not say that I think that quantum mechanics does fail at large distances, I only say that it is not inconsistent with what we do know. Ifthis failure of quantum mechanics is connected with gravity, we might speculatively expectthis to happen for masses such that GM ~ c = 1, of M near 10 − grams, which correspondsto some 10 particles ([6], pp. 12-13).The same problem is pointed out again later ([6], p. 14), where the possibility is put forward thatamplitudes may reduce to probabilities for a sufficiently complex object, as a consequence of a smearingeffect on the evolution of the phases of all parts of the object. Such a smearing effect could be relatedto the existence of gravitation. A similar idea is expressed in the end of the letter to Weisskopf [29],where he writeshow can we experimentally verify that these waves are quantized? Maybe they are not.Maybe gravity is a way that quantum mechanics fails at large distances.It is remarkable that, although Feynman often expressed his belief in the deeply quantum nature ofreality, including gravity, he was nevertheless open to the possibility that it is not so, until experimentsto clear the question are available. The possibility of a gravity induced collapse of wave function, as a solution to the measurement problemin quantum mechanics, can be traced back to the wide debate grown up at Chapel Hill about gravityquantization and Feynman’s deep insights. This is a reliable guess, because gravity is ubiquitousin all existing interactions, and gravitational effects depend on the size of objects; it later triggereda huge amount of investigations [44]-[53]. Among these, the Penrose proposal relies strongly on theconflict between the general covariance of general relativity and the quantum mechanical superpositionprinciple [50, 51]: it emerges when considering a balanced superposition of two separate wave packetsrepresenting two different positions of a massive object. If the mass M of the lump is large enough, thetwo wave packets represent two very different mass distributions. By assuming that in each space-timeone can use the notions of stationarity and energy, while the difference between the identified time-translation operators gives a measure of the ill-definiteness or uncertainty of the final superposition’senergy, the decay time for the balanced superposition of two lumps is: t D = ~ ∆ E grav . (6)15ere ∆ E grav is the gravitational self-energy of the difference between the mass distributions of each ofthe two lump locations. Thus it is clear that massive superpositions would never be formed becausethey would decay immediately. Later Penrose suggested that the basic stationary states into whicha superposition of such states decay are stationary solutions of the so called Newton-Schr¨odingerequation [52]. Since then, other collapse models (the so called dynamical collapse models) have beenalso suggested, due to a number of authors [54]-[57], where the collapse of the wave function of aquantum system is induced by the interaction with a random source, as for instance an externalnoise source. A lot of experimental proposals have been put forward as well, the challenge beingto explore a parameter regime where both quantum mechanics and gravity are significant. Thismeans that objects have to be massive enough to show gravity effects while still allowing a quantummechanical behavior. Currently it has been possible to build up a quantum superposition state withcomplex organic molecules of mass as large as of the order m = 10 − kg [58, 59]. Today prospectivetypical experiments, revealing gravity-induced decoherence, involve matter-wave interferometers [60]-[62] and quantum optomechanics [63]-[66] and magnetomechanics [67]. However, despite the hugetheoretical and experimental effort, the internal fundamental dichotomy between unitary deterministicquantum dynamics and nonlinear irreversible state collapse following a measurement process remainsstill unsolved, as well as the problem of the emergence of classical from the quantum world. Another hot topic emerged from discussions on quantum gravity, namely the role of observer in aclosed Universe, pointed out by Wheeler as follows:General relativity, however, includes the space as an integral part of the physics and it isimpossible to get outside of space to observe the physics. Another important thought isthat the concept of eigenstates of the total energy is meaningless for a closed Universe.However, there exists the proposal that there is one “universal wave function”. Thisfunction has already been discussed by Everett, and it might be easier to look for this“universal wave function” than to look for all the propagators ([1], p. 270).Here the first presentation is given of Everett’s relative-state interpretation of quantum mechanics[68]. Feynman promptly replies with his “many-worlds” characterization of Everett’s approach:The concept of a “universal wave function” has serious conceptual difficulties. This isso since this function must contain amplitudes for all possible worlds depending on allquantum-mechanical possibilities in the past and thus one is forced to believe in the equalreality of an infinity of possible worlds ([1], p. 270).The same idea will be expressed by Feynman some years later in his lectures on gravitation ([6], pp.13-14), where the role of the observer in quantum mechanics is also discussed by making explicitreference to Schr¨odinger’s cat paradox. In particular, an external observer is in a peculiar positionbecause he always describes the result of a measurement by an amplitude, while the system collapses16nto a well-defined final state after the measurement. On the other hand, according to an internalobserver, the result of the same measurement will be given by a probability. Thus a paradoxicalsituation emerges in quantum mechanics in the absence of an external observer, which also applieswhen considering the whole Universe as described by a complete wave function without an outsideobserver. The Universe wave function obeys a Schr¨odinger equation and implies the presence of aninfinite number of amplitudes, which bifurcate from each atomic event. For an inside observer, thiswould imply that he knows which branch the world has taken, so that he can follow the track of hispast. Feynman concludes the argument by raising a conceptual problem:Now, the philosophical question before us is, when we make an observation of our track inthe past, does the result of our observation become real in the same sense that the finalstate would be defined if an outside observer were to make the observation ([6], p. 14)?Later in the lectures on gravitation ([6], pp. 21-22), Feynman returns briefly to the meaning of thewave function of the Universe and confirm his “many-worlds” characterization of Everett’s approachby resorting to a cat paradox on a large scale , from which our world could be obtained by “reductionof the wave packet”. And he finally wonders on the mechanism of this reduction, the crucial issuebeing how to relate Everett’s approach and collapse mechanisms of whatever origin.Everett’s analysis [68] is, indeed, the first attempt to generalize the Copenhagen formulation inorder to apply quantum mechanics to the Universe as a whole. This requires to overcome the sharpseparation of the world into “observer” and “observed”, showing how an observer could become part ofthe system while measuring, recording or doing whatever operation according to usual quantum rules.Quantum fluctuations of space-time in the very early Universe have also to be properly taken intoaccount. On the other hand, it lacks an adequate description of the origin of the quasi-classical realmas well as a clear explanation of the meaning of the branching of the wave function. A further extensionand completion of Everett’s work has been developed by a number of authors [69]-[75] and is todayknown as decoherent histories quantum mechanics of closed systems . Within this formulation neitherobservers nor their measurements play a prominent role, and the so called retrodiction is allowed,namely the ability to construct a history of the evolution of the Universe towards its actual state byusing today’s data and an initial quantum state. As a matter of fact, retrodiction is fundamental forcosmology. Furthermore a rule is available to select out histories of the closed system under study(the Universe) to which probabilities can be assigned and to set a value for such probabilities. Thecriterion is to assign a probability to the sets of coarse grained alternative histories characterized bya negligible interference between the various histories in the set, corresponding to a particular initialstate of the closed system. In other words, the process of prediction requires to select out decoherentsets of histories of the Universe as a closed system, while decoherence in this context plays the samerole of a measurement within Copenhagen interpretation. Decoherence is a much more observer-independent concept and provides a clear meaning of Everett’s branches, the main issue being toidentify mechanisms responsible for it. In particular, it has been shown that decoherence is frequentin the Universe for histories of variables such as the center of mass positions of massive bodies [76].17
Gravity as a Quantum Field Theory
In a series of critical comments ([1], pp. 272-276), Feynman advocates a non-geometric and fieldtheoretical approach to gravity, which will later form the basis of the first part of his Caltech lectures[6]. He supposes that, in a parallel development of history, the general theory of relativity had notbeen discovered, but that the principles of lorentzian quantum field theory were known. Then, heasks, how would people deal with the discovery of a new force, namely the gravitational force? Theroots of such an approach can be traced back to Kraichnan [77] and Gupta [78] (more details aboutthis fascinating story can be found in [79]). Basically, one can use general arguments from field theoryand from experiment to infer that gravity (assumed to be mediated by a virtual particle exchange asall other forces) has to be carried by a massless spin-2 quantum, called graviton. Then full generalrelativity should follow from the principles of Lorentz invariant quantum field theory as applied to amassless neutral spin-2 field, as well as from consistency requirements. The same procedure, forthe spin-1 case, is well known to give Maxwell’s equations in the classical limit, and this approachis in line also with Feynman’s views about fundamental interactions, which he expressed in severalplaces [12]. In fact, as already emphasized in the introduction, according to Feynman the forces thatwe observe macroscopically, and that we dub as fundamental, must emerge from quantum theories inthe classical limit, and gravity is no exception.
In gravity, nonlinearity arises from the fact that the graviton has to couple universally with anythingcarrying energy-momentum, including itself, and general covariance along with the usual geometricinterpretation of general relativity come about merely as interesting and useful (albeit somewhatmysterious) byproducts, which can be seen as related to gauge invariance. In his words:The fact is that a spin-two field has this geometric interpretation; this is not somethingreadily explainable – it is just marvelous. The geometric interpretation is not really neces-sary or essential to physics. It might be that the whole coincidence might be understoodas representing some kind of gauge invariance ([6], p. 113).Feynman’s declared purpose, besides understanding another part of physics in his own and original way,was to trace a fast track towards the quantization of gravity, which was for him just the quantizationof another field. In particular, quantum gravity effects would be taken into account by includingdiagrams with closed loops. By treating gravity in this way, difficult conceptual and technical issuesconcerning the meaning of quantum geometry would not show up ([90], p. 377).In his critical comments, delivered at Chapel Hill conference ([1], pp. 272-276), Feynman startsfrom the fact that gravity behaves as a 1 /r force, through which like charges (i.e. masses) attract. The linear theory for such a field and its massive counterpart was completely worked out by Fierz and Pauli in Ref.[80], following a previous work by Dirac [81]. See, however, [82]. Other people who pursued the Lorentz invariant field theoretical approach to gravity include Weinberg [83, 84],Deser [85]-[88] and Wald [89].
18y trial and error one would then arrive at the hypothesis that it is mediated by a new massless spin-2field, which could be analogous to the spin-1 field of electrodynamics. Then the action would be: Z (cid:18) ∂A µ ∂x ν − ∂A ν ∂x µ (cid:19) d x + Z A µ j µ d x + m Z ˙ z µ d s + 12 Z T µν h µν d x + Z (second power of first derivatives of h ) , (7)where h µν is the new field, satisfying second order equations of the kind: h µν,σ,σ − h µσ,νσ = ¯ T µν . (8)Here the bar operation on a general second rank tensor X µν is defined as:¯ X µν = 12 ( X µν + X νµ ) − η µν X σσ . (9)The corresponding equation of motion for particles would be: g µν ¨ z ν = − [ ρσ, µ ] ˙ z ρ ˙ z σ , (10)where g µν = η µν + h µν , and [ ρσ, µ ] are the Christoffel symbols of the first kind: these equations arenothing but the linearized Einstein equations. By continuing the analogy with electromagnetism, acrucial fact here is that the Maxwell equations automatically imply that the current j µ is conserved;now the crucial issue is to find a suitable T µν such that the condition ∂ ν T µν = 0 is satisfied. However,a problem arises if particles move according to Eq.(10) or, what is the same, if the field h µν iscoupled to the matter, since the corresponding T µν does not obey to a conservation law: the lineartheory leads to a consistency problem. Also adding a further contribution t µν to the stress energytensor due to the gravitational field itself, thus replacing the term R T µν h µν d x in the action (7)with R ( T µν + t µν ) h µν d x , the problem is not overcome, since a variation of h would give new terms.According to Feynman, a working solution could be obtained, only by adding to the action a nonlinearthird order term in h µν , which gives for T µν the following equation: g µλ T µν,ν = − [ ρν, λ ] T ρν . (11)One could then proceed to the next higher order of approximation and so on but, as a matter of fact,finding the general solution of Eq. (11) is a very difficult task, which may be pursued by finding anexpression that is invariant under the following infinitesimal transformation of the tensor field g µν : g ′ µν = g µν + g µλ ∂A λ ∂x ν + g νλ ∂A λ ∂x µ + A λ ∂g µν ∂x λ . (12)where the 4 − vector A λ is the generator. This is a geometric transformation in a Riemannian space, The original equations appearing in [1] are schematic and present several index and sign mistakes; however, in noway this fact affects the reasoning we are reporting, and in the following we write the correct versions of such equations.
Feynman ends his comments with some speculations on a possible theory of gravitation built onalready known fields, i.e. without invoking a brand new spin-2 field, a promising candidate beingthe neutrino, described by a weakly coupled field with zero rest mass. But: how to get a force likegravitation by exchange of neutrinos? The exchange of one neutrino is ruled out by its half-integerspin, leading to orthogonal initial and final states orthogonal, and similarly a two neutrino exchangehas to be discarded because the resulting potential falls off faster than 1 /r . A possible solutionemerges from exchanging one neutrino between two bodies while, in turn, each body exchanging oneneutrino with the rest of the Universe located at a fixed distance. This situation, however, gives riseto a logarithmic divergence in the potential, and an higher order divergence would be obtained if onetries, for instance, four neutrino processes. Thus, Feynman is led to the conclusion that a theorybuilt of neutrinos doesn’t produce useful results: “This is obviously no serious theory and is not tobe believed” ([1], p. 276). The inadequacy of a theory of gravitation built with neutrinos would bediscussed in much more detail in the lectures on gravitation ([6], Lecture 2, pp. 23-28). After 1957, Feynman greatly developed the ideas on gravitation which he expressed at Chapel Hill,by giving a nearly complete derivation of Einstein’s equation from the spin-2 theory, discussing loopdiagrams, renormalization, unitarity, and considering the recent applications of the theory to astro-physics and cosmology. First of all, a preliminary assessment of his progress was given at the LaJolla conference in 1961 [93], followed by the already mentioned letter to Weisskopf [29], where thegravitational radiation is mainly discussed from the purely quantum viewpoint, noting that “withoutthe radiation correction there is no difficulty”. Such applications, indeed, had meanwhile begun to flourish, with the discovery of a wealth of phenomena, such asthe cosmic microwave background, pulsars and quasars (see e.g. [91] and references therein). The International Conference on the Theory of Weak and Strong Interactions was held in June 14-16 1961 at theUniversity of California, San Diego, in La Jolla. We recall that, here, G. F. Chew gave his celebrated talk on the S -matrix[92], while an afternoon session was devoted to the theory of gravitation, with Feynman reporting on his work on therenormalization of the gravitational field and recognizing nonunitarity as the main difficulty, shared also by Yang-Millstheory.
20 full discussion of gravitation was later given in his lectures on gravitation, delivered at Caltechin 1962-63 [6], while in the famous Warsaw talk Feynman introduced ghosts for properly handlingloop diagrams in gravity [2], thus fulfilling in part his goal to quantize gravity. Interestingly enough,the latter was the first paper that he published on gravity, despite an interest which at the timehad been going on for almost ten years. After that, Feynman only published two more papers onthe subject [94, 90], in the Festschrift for John Wheeler’s 60th birthday [95], apparently presentingresearch performed several years before that Feynman was reluctant to publish [79] due to its presumedincompleteness. For the same reason, the last Caltech lectures, addressing the quantization of gravity,were not included in Ref. [6]. Other discussions include the Hughes lectures on astronomy, astrophysicsand cosmology given in 1966-7 [10], where general relativity is treated (notably, this appears also inthe undergraduate Caltech lectures [96], but a traditional viewpoint is here taken).Both in the Caltech and in the Hughes lectures, Feynman gave some further justification for thechoice of a spin-2 field, based on the following observation. Unlike the electric charge, which is thesource of the electromagnetic force, the source of the gravitational force – energy – is not a relativisticinvariant, but rather grows with the velocity. Since, according to Feynman, the charge associatedwith a hypothetical scalar field would decrease with the velocity, and also due to the observationof light deflection, the possibility of a (spin-0) scalar field is ruled out. Feynman thus singles outthe spin-2 case and, in the Caltech lectures, he right away starts constructing a quantum theory forsuch a field. Only after completing the derivation of the Einstein field equations, which takes severallectures, he links his approach with the usual one by stating the principle of equivalence and explainingthe geometrical interpretation. After that, he switches to applications, most notably considering theSchwarzschild solution, wormholes, black holes, supermassive stars and cosmology, then closing withthe already cited lecture on gravitational radiation.In the Hughes lectures instead, after commenting more about the justification for the choice of aspin-2 field, he switches directly to the usual, geometric approach, probably because the field theoret-ical derivation would be both too advanced for the audience, and too long for a course whose focuswas on applications.
The issue of loop corrections and renormalization in quantum gravity is publicly addressed by Feyn-man, for the first time (to the best of our knowledge), in 1961, at the Conference in La Jolla [93]. Herehe recognizes nonlinearity as a source of difficulty for both gravitation and Yang-Mills theory, the twotheories sharing similar nonlinear equations for the quantum field. Basically, gravitational sources areenergy and momentum, which are locally conserved, and the gravitational field carries energy andmomentum itself, so it is self-coupled. Similarly, the source of a Yang-Mills field is the isotopic spincurrent, which is as well locally conserved, and the Yang-Mills field carries isotopic spin itself, andthus it is also self-coupled. In both cases, the result is a nonlinear field theory. Starting from the late ’60s, in fact, Feynman became more and more absorbed in studying partons and stronginteractions, which were his main interest in the ’70s. .1 Attacking the problem An up to date contribution to the subject is contained in Ref. [2], a report of the talk given at thethird General Relativity conference (GR3), held at Warsaw in 1962 , and in two papers, writtensome years later, for the Festschrift to celebrate John Wheeler’s 60th birthday [94, 90]. Accordingto his original strategy, Feynman does not follow the usual approach based on the quantization ofspace-time geometry, but rather prefers to construct a quantum field theory for a massless spin-2 field– the graviton – and then work out the results at different perturbative orders. As we have discussedabove, the tree-diagram approximation gives the classical limit, i.e. the Einstein equations, whileloop diagrams describe quantum corrections. Unlike lectures [6], here the Einstein equations and thecorresponding action are assumed from the beginning, and then quantized by adopting a standardprocedure.At the starting of the Warsaw paper [2] Feynman vividly outlines his aims and approach onceagain:My subject is the quantum theory of gravitation. My interest in it is primarily in therelation of one part of nature to another. There’s a certain irrationality to any work ingravitation, so it’s hard to explain why you do any of it; for example, as far as quantumeffects are concerned let us consider the effect of the gravitational attraction between anelectron and a proton in a hydrogen atom; it changes the energy a little bit. Changing theenergy of a quantum system means that the phase of the wave function is slowly shiftedrelative to what it would have been were no perturbation present. The effect of gravitationon the hydrogen atom is to shift the phase by 43 seconds of phase in every hundred timesthe lifetime of the Universe! [...] I am limiting myself to not discussing the questionsof quantum geometry nor what happens when the fields are of very short wave length.[...] I suppose that no wave lengths are shorter than one-millionth of the Compton wavelength of a proton, and therefore it is legitimate to analyze everything in perturbationapproximation; and I will carry out the perturbation approximation as far as I can inevery direction, so that we can have as many terms as we want ([2], p. 697).As in his Caltech lectures, here Feynman points to a substantial unity of nature, which requires toreconcile gravity and quantum mechanics, although he admits that the work has some irrationalityin it due to the smallness of the effects. Consistently with what he said at Chapel Hill concerningthe choice between mathematical rigor and thought experiments, he chooses the latter, then pursuinga perturbative approach by solving a simple specific problem and later switching to a more complexone: So please appreciate that the plan of the attack is a succession of increasingly complexphysical problems; if I could do one, then I was finished, and I went to a harder one Interestingly, some notes that apparently refer to the Warsaw talk are included in Feynman’s notes for the famousCaltech lectures, which were given in the same period. In particular, in Folder 40.5, which is available online [97], thelast two pages specifically discuss gravity, Yang-Mills theory and loop corrections, with the heading “Grav talk”. Thisfurther shows how deeply involved in these matters he was in that period.
In Feynman’s words, the starting point is the Einstein Lagrangian for gravity:I started with the Lagrangian of Einstein for the interacting field of gravity and I had tomake some definition for the matter since I’m dealing with real bodies and make up mymind what the matter was made of; and then later I would check whether the results thatI have depend on the specific choice or they are more powerful ([2], p. 698).His approximation for the metric is the following: g µν = δ µν + κh µν , (13)which allows, upon substitution and subsequent expansion, to cast the lagrangian in the form: L = Z (cid:0) h µν,σ ¯ h µν,σ − h µσ,σ ¯ h µσ,σ (cid:1) d τ + 12 Z (cid:0) φ ,µ − m φ (cid:1) d τ + κ Z (cid:18) ¯ h µν φ ,µ φ ,ν − m h σσ φ (cid:19) d τ + κ Z ( hhh ) d τ + κ Z ( hhφφ ) d τ + ... (14)where the simplified notation ¯ h µν = ( h µν + h νµ − δ µν h σσ ) has been introduced, and a schematicnotation has been adopted for the highly complex higher terms. Obviously one recognizes in the firsttwo terms the free Lagrangian of the gravitational field and of matter.The first step is to try to solve the problem classically, which is carried out by varying the La-grangian (14) with respect to h and, then, to φ . The following equations of motion with a source termare obtained: h µν,σσ − ¯ h σν,σµ − ¯ h σµ,σν = ¯ S µν ( h, φ ) , (15) φ ,σσ − m φ = χ ( φ, h ) . (16)The strategy is to solve the equations by using quantum theory methods and by following a proce-dure analogous to electromagnetism in order to get the propagators. But soon he realizes that Eq.(15) is singular, so he proceeds by resorting to the invariance of the Lagrangian under the followingtransformation: h ′ µν = h µν + 2 ξ µ,ν + 2 h µσ ξ σ,ν + ξ σ h µν,σ , (17)where ξ µ is arbitrary. As a consequence, the consistency of Eq. (15) requires the source S µν tohave zero divergence. By making also the simple gauge choice ¯ h µσ,σ = 0, he finally gets the law of23ravitational interaction of two systems by means of the exchange of a virtual graviton, although heobserves that, in order to obtain an higher accuracy, one has to work out radiative corrections. Besidesworking out the propagator, Feynman gives other examples of calculation with diagrams by explicitlycomputing the amplitude for the coupling of two particles to a graviton, namely an interaction vertex,and finally considers the analogue of the Compton effect, where the photon is replaced by a graviton. Feynman then switches to a more complex situation, which requires to go beyond the tree levelapproximation:However the next step is to take situations in which we have what we call closed loops, orrings, or circuits, in which not all momenta of the problem are defined ([2], pp. 703-704).However, the one-loop approximation brings into game a bunch of conceptual issues, as clearly ex-pressed by Feynman’s words:This made me investigate the entire subject in great detail to find out what the trouble is.I discovered in the process two things. First, I discovered a number of theorems, which asfar as I know are new, which relate closed loop diagrams and diagrams without closed loopdiagrams (I shall call the latter diagrams “trees”). The unitarity relation which I have justbeen describing, is one connection between a closed loop diagram and a tree; but I founda whole lot of other ones, and this gives me more tests on my machinery. So let me justtell you a little bit about this theorem, which gives other rules. It is rather interesting.As a matter of fact, I proved that if you have a diagram with rings in it there are enoughtheorems altogether, so that you can express any diagram with circuits completely in termsof diagrams with trees and with all momenta for tree diagrams in physically attainableregions and on the mass shell. The demonstration is remarkably easy ([2], p. 705).By working out in detail one loop calculations, Feynman soon realizes that unitarity gets lost becausesome contributions arise from longitudinal polarization states of the graviton, which don’t cancel. Assuggested by Gell-Mann, he worked out a similar problem in a simpler context, that of masslessYang-Mills theory, and found the same pathologic behavior:But this disease which I discovered here is a disease which exist in other theories. Soat least there is one good thing: gravity isn’t alone in this difficulty. This observationthat Yang-Mills was also in trouble was of very great advantage to me. [...] the Yang-Mills theory is enormously easier to compute with than the gravity theory, and therefore Icontinued most of my investigations on the Yang-Mills theory, with the idea, if I ever curethat one, I’ll turn around and cure the other ([2], p. 707). This circumstance is also mentioned in Ref. [4].
Conclusive remarks about quantum correction deal with the possibility of extending to higher orders(e.g. two or more loops) the above results:Now, the next question is, what happens when there are two or more loops? Since I only gotthis completely straightened out a week before I came here, I haven’t had time to investigatethe case of 2 or more loops to my own satisfaction. The preliminary investigations that Ihave made do not indicate that it’s going to be possible so easily gather the things intothe right barrels. It’s surprising, I can’t understand it; when you gather the trees intoprocesses, there seems to be some loose trees, extra trees ([2], p. 710).Clearly, he doesn’t know how to manage with this stuff and the same feeling is expressed in his lectureson gravitation:I do not know whether it will be possible to develop a cure for treating the multi-ringdiagrams. I suspect not – in other words, I suspect that the theory is not renormalizable.Whether it is a truly significant objection to a theory, to say that it is not renormalizable,I don’t know ([6], Lecture 16, pp. 211-212).In the same lecture, Feynman also argues for a substantial non-renormalizability of the theory.25urther clarifying remarks on the tree theorem, on the proof of this theorem in the one loopcase and on the nature of the fictitious particle, needed in order to guarantee unitarity, are given byFeynman in the subsequent discussion, when answering to a question by DeWitt. The same issuesare developed in more detail in the already mentioned couple of papers, written for the Wheeler’sFestschrift [94, 90], giving many more details. In particular, in Ref. [94], the theorem connecting treesand loops is treated in detail, while a large part of Ref. [90], is devoted to the previously unpublishedresults Feynman got on Yang-Mills theory.Later, inspired by Feynman’s talk in Warsaw and by the discussion that followed, DeWitt solvedthe problem of extending Feynman’s formalism to two [98] and arbitrarily many [99, 100] loops.However, while Yang-Mills theory was later shown to be renormalizable [101]-[104], the divergencesof gravity proved to be too strong to be tamed by renormalization [105]-[108], confirming Feynman’ssuspect. It is noteworthy that, against the common lore of the time, Feynman was not convinced thatnon-renormalizability meant that a theory was inconsistent . As he himself declared in one his lastinterviews, given in January 1988 (quoted in [109]):The fact that the theory has infinities never bothered me quite so much as it bother others,because I always thought that it just meant that we’ve gone too far: that when we go tovery short distances the world is very different; geometry, or whatever it is, is different,and it’s all very subtle.Modern views on quantum field theory, emerged much later, indeed see non-renormalizability to benot a significant objection against a theory, but rather as a signal that the theory loses validity atenergies higher than a certain scale, i.e. it is an effective field theory , which can however be used tomake predictions under that scale. As it is now well understood, this is just the case of gravity (seee.g. [110] and references therein). It is also of interest to notice that Feynman’s tree theorem hasrecently been considered by physicists working in the field of advanced perturbative calculations andgeneralized unitarity (see e.g. [111]-[115]). Some other, interesting documents reveal further elaboration by Feynman on the subject. In thefollowing we briefly discuss them in order to better understand his attitude and arguments.
The lectures [10] that Feynman gave in 1966-67 at the Hughes Aircraft Company are of greatinterest not only as a further example of Feynman’s curiosity in action in a different field, but also, forwhat concerns the present work, because they contribute to our understanding of Feynman’s vision of This is also recalled by Gell-Mann in Ref. [4], where he states that “he was always very suspicious of unrenormaliz-ability as a criterion for rejecting theories”. Like other sets of lectures given in those years, the presently analyzed ones are available to us thanks to the effortsof John T. Neer, who also undertook the task of including some up to date information. . The remaining chapters,and Feynman’s involvement in applied gravitation matters like astrophysics and cosmology, will bethe subject of a forthcoming publication.The lectures begin with an overview of the subject matter, organized by growing scale, i.e. fromthe solar system to quasars. After that, the first topic addressed by Feynman is the Universe as awhole, namely cosmology. This is the subject of the last part of chapter 1 and of chapter 2 of the notes,in which a model Universe treated using Newtonian gravity is discussed. Then, in chapter 3, Feynmanstarts discussing general relativity, by despising the geometric approach (cf. the quotation we reportedin Sec. 2) and giving some motivation about why gravity should be described by a tensor field, alongthe lines of the Caltech lectures [6]. After that, however, as already remarked, he switches to theusual, Einsteinian views and, very interestingly, for each step he rightly credits Einstein for taking it,but often he gives a quite original motivation. He then begins explaining the different notions of mass,the equivalence principle (stated by the usual elevator thought experiment) and gravitational redshift,later including an interesting comment of the well known problem of the radiation by a freely fallingcharge. This same issue was used by him in Ref. [6] (pp. 123-4) to argue in favor of the modificationof electromagnetism in a gravitational field; this was a hot topic in the first half of the sixties, studiedby people like Rohrlich [116], DeWitt [117] and others.In a very interesting discussion ([10], pp. 38), Feynman uses a variational approach to show thata traveling clock maximizes the time elapsed, if it satisfies the usual Newtonian equation of motionof a particle subject to the gravitational acceleration g , i.e. if it is freely falling. Such a discussionwas also present in the Caltech lectures, but no such calculation was performed there, and this simplecomputation is used as a motivation for the relativity of time in a gravitational field, as well as for theintroduction of a metric field in the expression of the usual proper time interval in order to generalizethis one, and for the geodesic equation. This is followed by the curved space interpretation of themetric, a discussion of curved geometry and a description of the Schwarzschild solution of the fieldequations. After a more lengthy description of the maximum proper time principle to find trajectories,this is applied to a very detailed discussion of the geodesics in the Schwarzschild solution, in order tofind the deflection of light due to the Sun. By contrast, in Ref. [6], such a computation is performedin the perturbative approach (pp.59-61), and repeated for the full Schwarzschild metric only for radialgeodesics (p. 201). Interesting, light bending is not found by discussing null geodesics, but ratherdiscussing the time-like geodesics – for a massive particle – and then taking its speed to be v = c . Theresult obtained is thus half the correct one, and Feynman does not make the correct computation,but rather hints to the fact that it can be obtained by considering the fact that curved space can beseen as a medium with a varying index of refraction (in [6], p. 41, he explains the factor of 2 as dueto magnetic-like gravitational interactions, which become equally important as the static ones for anobject that moves with the speed of light). A qualitative discussion of the Schwarzschild radius and Notice that the chapter numeration in the table of contents at the beginning of the lectures does not match thenumeration in the main text. Here we refer to the latter.
27f black holes (or “black stars”, as appropriate for those times) then follows.
While motivating some steps in [10], Feynman inserts some statements that may hint to the fact thathe had been thinking at gravity in a way analogous to his treatment of electromagnetism, that wasrecently uncovered in [118] and in [13], and discussed in [11] and [12] (to which we refer for details).As we briefly recall, Feynman developed a formulation of electromagnetism that was relativisticfrom scratch, and having its roots in his belief that nature is fundamentally quantum, with classicalfundamental interactions emerging from quantum theories: these points are at the basis also of hisviews of gravity, as we repeatedly emphasized. His treatment starts by noticing that, for electro-magnetism, the relevant charge is unchanged by motion, and giving prominence to electromagneticpotentials, since they are the basic objects in the quantum theory. For gravity, we have already seenhow a similar reasoning led him to the conclusion that the relevant potential for gravity had to be atensor, and, differently from electromagnetism, general relativity is always formulated by starting fromthe potentials (i.e. the metric), while the analogs of the electric and magnetic fields, (the connectioncoefficients) are considered as derived objects. Thus, an approach to gravity analogous to Feynman’sapproach to electromagnetism is surely something that makes sense.As described in [118, 11], Feynman used relativity to argue that the force felt by an electric chargein an electromagnetic field has to be linear in the velocity. Then, by imposing relativistic covariance, hemanaged to restrict the force to assume the form of a Lorentz force, and to derive the transformationproperties of the coefficients. For gravity, Feynman argues that the force acting on a massive particlehas to be quadratic in the velocity, rather than linear, due to the higher tensor character of gravitationwith respect to electromagnetism. Thus the general expression is ([10], p. 33): F i = m ( C i + v j β ij + v j v k δ ijk ) , i, j, k = 1 , , , (18)where C i is the usual Newtonian field, and m is the (gravitational) mass of the given particle.Presumably, this discussion should motivate – in Feynman’s aims – why the geodesic equation isquadratic in the velocity of the moving particle, as he states:What Einstein did was then to set out to find the laws of motion and the laws determiningthe coefficients.For electromagnetism, after deriving the Lorentz force, Feynman then switched to the derivation ofMaxwell’s equations; this is developed in Ref. [13], and contains several original points. In particular,the homogeneous Maxwell equations are obtained from the observation that an invariant action fora relativistic particle in a field can be written by adding to the free action an invariant 4-potentialterm. Remarkably, in [13] (p. 42) there is also a hint of a possible extension to gravity, by noticingthat 10-potential could be added as well. In such a case, the action should read: S = − m c Z (cid:18) d τ + 12 c h µν d x µ d τ d x ν d τ d τ (cid:19) , (19)28nd a comment follows: “An example of a force derivable from that action is gravity.” It has to benoticed that Feynman replaced h with g in the above action, although we have found that the correctexpression is just what reported above. Also, we have included a 1 / h µν to be identified with the usual perturbations of the metric.We further notice that the m factor appearing in front of the action is the inertial mass, which isof course the same as the gravitational mass appearing in (18). Feynman, however, did not performthose computations, likely because they would have been very cumbersome. In this paper we have reviewed Feynman’s picture of gravity by considering all known publishedsources, as well as many unpublished ones, including the Hughes lectures [13], which were madeavailable only recently. As in all fields in which Feynman was engaged, his deepness and originalitycan be seen everywhere.The main feature emerging from our discussion is his belief in the innermost unity of nature,which, at its deepest level, has to be quantum. This view is reflected in the statement that thefundamental interactions that we experience at the macroscopic level are manifestations of underlyingquantum theories, and general relativity can be indeed obtained (as the Maxwell theory) just fromthe fundamental principles of lorentzian field theory. This approach also gives a recipe for consideringquantum corrections, and we saw how Feynman dealt with the emerging difficulties. Usually Feynmanstarted from this viewpoint to make even classical computations in gravity, such as the orbits of planetsin Ref. [6] and the radiation of gravitational waves in ref. [29], but the Hughes lectures [13] (and alsothe final chapters of volume II of Ref. [96]) prove that he could also put aside that approach in favorof a more conventional one if the audience required it. Even then, however, his originality emergesclearly.We cannot avoid being awed by the vastness and deepness of Feynman’s contributions to physics,which continues to unfold in front of our eyes, each time we dig a new little piece of it, without riskof being disappointed.
Acknowledgments
The authors would like to thank the staff of the Caltech archives for providing them a copy of Feyn-man’s letter to Weisskopf [29].
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