Eikonal Analysis of LinearizedQuantum GravityA Functional ApproachbyColby L. DeLisleB.Sc., University of Missouri, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Colby L. DeLisle 2017AbstractThe low energy effective theory of quantized gravity is currently our mostsuccessful attempt at unifying general relativity and quantum mechanics. Itis expected to serve as the universal low energy limit of any future micro-scopic theory of quantum gravitation, so it is crucial to properly understandits low frequency, long wavelength, “infrared” limit. However, this effec-tive theory suffers from the same kind of infrared divergences as theorieslike quantum electrodynamics. It is the aim of this work to characterizethese divergences and isolate the infrared behavior of quantum gravity us-ing functional methods. We begin with a review of infrared divergences, andhow they are treated in QED. This includes a brief overview of the knownapplications of functional methods to the problem. We then discuss the con-struction of the effective field theory of quantum gravity in the linearizedlimit, coupled to scalar matter. Proceeding to the main body of the work,we employ functional techniques to derive the form of the scalar propaga-tor after soft graviton radiation is integrated out. An eikonal form for thegenerating functional of the theory is then presented. In the final chapter,we use this generating functional to derive the soft graviton theorem andthe eikonal form of the two-body scalar scattering amplitude. The result isa concise derivation of multiple known results, as well as a demonstrationof the factorization of soft graviton radiation against the eikonal amplitude.We conclude with some comments on how these results can be extended, andwe argue that the functional framework is the best candidate for a unifiedunderstanding of all relevant infrared features of quantum gravity.iiLay SummaryIt is often said that the two most successful theories of modern physics -quantum mechanics and general relativity - are incompatible. This is notstrictly true. At low energies, these two theories can be combined to somesuccess, though the approach has its own limits. This work examines howto unify the discussion of various phenomena in quantum gravity under onemathematical framework, in the limit of extremely low energies. It is hopedthat such a description will also contain insight into more general propertiesof quantum gravity and quantum field theory.iiiPrefaceThis dissertation is original, unpublished, independent work by the author,C. DeLisle.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Infrared Problems in Quantum Field Theory . . . . . . . . . 41.1.1 IR Divergences . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Functional Methods . . . . . . . . . . . . . . . . . . . 111.2 Theory of Quantized Gravity and Matter Fields . . . . . . . 141.2.1 The Lagrangian Formulation of GR . . . . . . . . . . 151.2.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . 161.2.3 Gravitons . . . . . . . . . . . . . . . . . . . . . . . . . 172 Eikonal Methods for Linearized Gravity . . . . . . . . . . . 222.1 Equation of Motion for the Scalar Field . . . . . . . . . . . . 232.2 The Generating Functional . . . . . . . . . . . . . . . . . . . 242.3 Propagator on a Fixed Background . . . . . . . . . . . . . . 26vTable of Contents2.4 Applying the Interaction Operators . . . . . . . . . . . . . . 292.5 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.1 The Two-Point Function . . . . . . . . . . . . . . . . 322.5.2 The Four-Point Function . . . . . . . . . . . . . . . . 332.6 Infrared Behavior of the Two-Point Function . . . . . . . . . 343 Amplitudes and Soft Theorems . . . . . . . . . . . . . . . . . 363.1 Gravitational Bremsstrahlung . . . . . . . . . . . . . . . . . 373.1.1 Constructing the Amplitude . . . . . . . . . . . . . . 373.1.2 Functional Eikonal Limit . . . . . . . . . . . . . . . . 383.1.3 On-Shell Results . . . . . . . . . . . . . . . . . . . . . 393.1.4 Multiple Graviton Emission . . . . . . . . . . . . . . 403.1.5 Virtual Graviton Exchange . . . . . . . . . . . . . . . 413.2 Two-Body Scattering . . . . . . . . . . . . . . . . . . . . . . 423.2.1 Constructing the Eikonal Amplitude . . . . . . . . . . 423.2.2 On-Shell Results . . . . . . . . . . . . . . . . . . . . . 443.2.3 Multiple Graviton Emission . . . . . . . . . . . . . . 453.2.4 Virtual Graviton Exchange . . . . . . . . . . . . . . . 453.2.5 Soft Factorization . . . . . . . . . . . . . . . . . . . . 474 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52AppendicesA Selected Functional Identities . . . . . . . . . . . . . . . . . . 58B Evaluation of Phase Integrals . . . . . . . . . . . . . . . . . . 60C The Generating Functional . . . . . . . . . . . . . . . . . . . . 62D The Generating Functional Generates the S-Matrix . . . . 64viList of Figures1.1 QED vertex loop correction . . . . . . . . . . . . . . . . . . . 51.2 Soft photon emission . . . . . . . . . . . . . . . . . . . . . . . 61.3 Graviton-matter vertex . . . . . . . . . . . . . . . . . . . . . 202.1 Loop correction to graviton polarization . . . . . . . . . . . . 23viiAcknowledgementsThe author would like to thank Philip Stamp for supervision and advice.Additional thanks go to Dan Carney, Tim Cox, Michael Desrochers, andJordan Wilson for many invaluable conversations.Sincere gratitude is owed to Kerry DeLisle, Clarke DeLisle, and ViktorijaJuciu¯te˙ for their personal support, without which this work would not havebeen possible.viiiFor Cindi.ixChapter 1IntroductionFrom Newtonian gravity to electroweak theory, the draw of unifying seem-ingly unrelated phenomena under one framework has been responsible formuch of the current landscape of physical understanding. Tying togetherquantum mechanics (QM) and general relativity (GR) remains perhaps themost sought-after unification in the field. In many aspects, the two theoriesseem utterly at odds with one another, and even the question of whethergravity must be quantized is still unanswered. However, it is a remarkablefact that assuming gravity is quantized, GR and QM can get along at lowenergies as an effective field theory. Indeed, in the low energy limit weexpect any sensible theory of quantized gravity to reduce to this effectivedescription in this picture. Using this intuition as motivation, we note thatthe infrared limit of the effective field theory may provide a more accessi-ble way to search for hints of a full unified theory of quantum gravity, asopposed to the more difficult path of quantizing GR at extremely high en-ergies and short distances, as is attempted by theories of strings [6], loops[7], etc.The problem with this approach is that even our best understood physi-cal theory - quantum electrodynamics - is still suffering from a problem ofinfinities in its infrared limit. While it is known in a technical sense howto make physical predictions by carefully discarding these divergences [29–31], it should be uncontroversial to say that their origin and cancellation isunsatisfactorily understood on physical grounds. The same can be said foranalogous problems that remain in the quantum gravity case.It is the purpose of this work to apply a powerful functional technique orig-1Chapter 1. Introductioninally developed by E.S. Fradkin [2] to the problem of IR divergences inquantum gravity. We aim to show that this method is well-suited to dis-cussion of these issues in a single language by reproducing known resultsfrom the literature. We also offer comments and examples of ways in whichthese known results can be extended or better understood in the functionalpicture. This work too is a sort of unification, in that we believe the func-tional picture of soft gravity provides an easier way to analyze all of thecontributions to IR issues in parallel.The organization of the thesis is as follows. In Chapter 1, we introduce thesegeneral issues arising in the study of IR physics in quantum field theory, us-ing QED as an example, and lay out the basic elements of the most simpletheory of quantum gravity coupled to matter - Einstein GR minimally cou-pled to one real scalar field. Chapter 2 then deals with how to analyze thistheory in the functional eikonal framework. We derive the eikonal form ofthe generating functional, as well as the scalar propagator. Finally, chapter5 shows how to use these results to compute S-matrix elements, as well asdemonstrating how the soft graviton theorem emerges naturally in this limitfrom the nonperturbative calculation. We conclude with some comments onwhat new results may emerge from this framework.What is meant by soft? For the entirety of this thesis, we define softradiation as that which has momentum q that satisfiesq2 m2, (1.1)where m is the mass of the scalar particle. For a scalar particle interactingwith virtual soft quanta, this means that the scalar loses (gains) negligiblemomentum to emitted (absorbed) gravitons. This enforces the conditionthat, throughout the process, the scalar remains nearly on-shell. Calling themomentum of the scalar at some point in the process k, this meansk2 ≈ m2, (1.2)2Chapter 1. Introductioneven in situations where the virtual particle momenta are integrated over.Furthermore, the words “soft” and “eikonal” will often be used interchange-ably.Regarding Notation: In this thesis, we use the mostly-minus metricsignature,ηµν = diag(+1,−1,−1,−1), (1.3)use commas to denote partial derivatives (A,α≡ ∂αA), and employ the Ein-stein summation convention, where repeated indices are summed implicitly.We also commonly suppress coordinate-dependence, Lorentz indices, inte-gration measures, and integration variables, where these things should beclear from the context. A few examples:∫fj ≡∫d4x f(x)j(x), (1.4)∫jA ≡∫d4x jµ(x)Aµ(x), (1.5)∫δδI∆δδI≡∫d4x∫d4yδδIµν(x)∆µναβ(x, y)δδIαβ(y), (1.6)etc. Unless otherwise specified, seeing∫with no measure implies∫d4x or∫d4x∫d4y depending on the context. For momentum integrals, we use thenotation(D)∑q≡∫dDq(2pi)D, (1.7)with the special case ∑q≡(4)∑q≡∫d4q(2pi)4, (1.8)since we usually work in (3+1) dimensions. Lastly, we choose to use naturalunits, where ~ = c = 1.31.1. Infrared Problems in Quantum Field Theory1.1 Infrared Problems in Quantum FieldTheoryQuantum field theory - quantum electrodynamics (QED) in particular - isthe most well-tested theory in the history of science [25]. It is the foundationof our current understanding of the interaction of matter via all forces exceptgravity, though as we will see, even gravity can in some limits be accountedfor. In spite of its experimental success, the theory is still imperfect. Wemore or less know how to obtain sensible results from QED calculationsdespite many integrals resulting in infinite answers, though some of thecurrent methods for dealing with such divergences leave a lot to be desired.Divergences that come from low energy massless excitations are particularlyinteresting, and in this section we will discuss why that is.In this introduction, we introduce the topic of IR divergences using resultsas presented in [29] and chapter 6 of [26], and borrow their conventions asneeded. The discussion of IR divergences is first presented from a diagram-matic perspective, followed by some background on the functional methodsused to attack the same problems. We assume the reader is familiar withQED and its Feynman rules.1.1.1 IR DivergencesConsider the amplitude corresponding to Figure 1.1 in QED.The momentum q can be thought of as an arbitrary external source. Evalua-tion of this diagram involves an integral over the virtual photon momentuml - an integral (eq. (6.38) of [26]) which is seen to diverge both as |l| → ∞and as |l| → 0. The first type of divergence, we call a UV divergence. Theseare infinities coming from the naive assumption that the theory of QED isvalid up to arbitrarily high energy scales. We will not be concerned withthis type of infinity in this work except to refer to textbook discussions ofrenormalization, in e.g. [5, 26].41.1. Infrared Problems in Quantum Field Theoryqlpp′Figure 1.1: Loop correction to the QED vertex.As the photon momentum goes to zero however, we encounter IR diver-gences. These are caused by the photon’s masslessness and we will discusstheir effects here. Beyond showing up as an infinity in the calculation of anamplitude, IR divergences appear in computation of observables. The effectof this diagram’s IR divergence on the differential cross section in the limitof strong external field (q2 →∞, used for computational convenience) turnsout to bedσdΩ(p→ p′) =(dσdΩ)0[1− αpiln(−q2m2)ln(−q2µ2)+O(α2)](1.9)where α ≡ e2/4pi, ( dσdΩ)0 is the cross section without the inclusion of thevirtual photon, m is the mass of the electron, and µ is a fictitious photonmass used to regulate the integral. Taking the massless limit, we see thatthis makes the logarithm diverge. This form was first discovered by Sudakov[27] and is now referred to as the Sudakov double logarithm.What are we to do with this? Of course observed cross sections do notdiverge. The key actually lies in the expression of a seemingly unrelatedprocess - the emission of a single low energy photon. This amplitude reallyhas two contributions, displayed in figure 1.2, as the photon can couple toeither the ingoing or outgoing fermion leg.This amplitude is tree level, and so easy to compute. The addition of the51.1. Infrared Problems in Quantum Field Theoryqpl +p′qpp′lFigure 1.2: Amplitude for scattering and producing a low energy photon.external photon, as |l| → 0, causes the following amendment to the originalcross section:dσdΩ(p→ p′ + γ) =(dσdΩ)0[αpiln(−q2m2)ln(−q2µ2)+O(α2)](1.10)Remarkably, this looks identical to the second term in (1.9), with the oppo-site sign. This means that these processes on their own seem to be unphys-ical, while their sum is perfectly finite. To interpret this result, we mustrealize that in any real experiment, we cannot measure emitted photons ofarbitrarily low energy. We denote the lower end of the momentum scale wecan resolve by λ. To compute the inclusive cross section of these two pro-cesses, we simply add the two previous results, but integrate the momentumof the emitted photon from 0 to λ, allowing for all photon energies thatare indistinguishable to our experiment. The divergent contributions to thephysical result givedσdΩphys=(dσdΩ)0[1− αpiln(−q2m2)ln(−q2λ2)+O(α2)]. (1.11)This is a result depending only on perfectly well-defined physical parameters.The continuation of this argument to higher orders in perturbation theoryis not trivial. Details can be found in [29, 31], but we will give an overviewof the results here.61.1. Infrared Problems in Quantum Field TheoryIn ref. [27], Sudakov actually goes beyond showing eq. (1.9). He sums allsimilar diagrams in perturbation theory (vertex corrections involving lowenergy virtual quanta) and shows that the result exponentiates. This meansthat the full contribution of the vertex correction, after summing to allorders, isdσdΩ(p→ p′) =(dσdΩ)0exp[−αpiln(−q2m2)ln(−q2µ2)+O(α2)]. (1.12)Notice that nowdσdΩ(p→ p′) µ→0−−−→ 0 (1.13)so that this IR divergence causes the cross section to vanish! What remainsis to look at the contribution of processes that involve emission of more thanone soft photon. We will go over this in some detail, as we will make contactwith this derivation in later chapters.Consider some known amplitude in QED. Let this amplitude be denotedM. Specifically, we consider the amplitude for n fermions to scatter to mfermions, with no incoming or outgoing photons.M =p′mp′1p1 pn(1.14)We now ask the question: what is the effect on this amplitude of adding oneadditional soft outgoing photon line to this amplitude? We take this photonto have momentum q, polarization µ(q), and assume the limit |q| → 0.71.1. Infrared Problems in Quantum Field TheoryM→M′ =qp′mp′1p1 pn(1.15)First we note that attaching the line anywhere inside the blob will not leadto any infrared divergences, as the particles in the blob are generically off-shell. However, consider what happens when we attach the line to the ithoutgoing on-shell matter line with momentum p′i.This multipliesM by a vertex function and a propagator factor. We enforcethe momentum shell condition for p′i, and neglect the small q2 terms, makingthe total multiplicative factore(p′i)µp′i · q − iδ, (1.16)which is clearly divergent as |q| → 0. Similarly, if we had attached thephoton line to the jth incoming fermion, we would multiply the amplitudeby a factor− e(pj)µpj · q + iδ . (1.17)For n→ m scattering then, the amplitude becomes simplyM′ =M× e m∑i(p′i)µp′i · q − iδ−n∑j(pj)µpj · q + iδ ≡M× eΩµν , (1.18)or in pictures:81.1. Infrared Problems in Quantum Field Theoryqp′mp′1p1 pn=p′mp′1p1 pn× Ωq(1.19)The important thing here is that the contribution from the soft radiationfactorizes against the known hard amplitude M. The omegas are factorsthat are gauge invariant and do not depend on the spin of the hard particles.The Ω notation is simply to have a consistent name for these soft factorsthroughout this work. It is also shown in [29] that the generalization to Nphoton emissions is straightforward:M(N) =M×N∏le m∑i(p′i)µp′i · ql − iδ−n∑j(pj)µpj · ql + iδ ≡M× N∏le(Ωµ)l,(1.20)or, attaching the outgoing photon wavefunction and normalization factorsto write the full S-matrix element:M×N∏le(2pi)3/2√2|ql| m∑i∗ · p′ip′i · ql − iδ−n∑j∗ · pjpj · ql + iδ ≡M× N∏leΩl.(1.21)Equation (1.21) is called the soft photon theorem, as it describes the changein the amplitude due to the addition of N external soft photon lines.Let us apply this result to our simple scattering example, generalizing theprocess in Figure 1.2. We recall that q now means p′ − p once again. Thecorrection to the cross section from all possible numbers of emitted pho-tons with unmeasurable momenta (also summed over polarizations) is given91.1. Infrared Problems in Quantum Field Theorybylimµ→0∑n1n!∫ λµdk02pi(3)∑k−e22k0(p′νp′ · k −pνp · k)(p′νp′ · k −pνp · k)n= exp[αpiln(−q2m2)ln(λ2µ2)]µ→0−−−→∞.(1.22)This infinite result seems unphysical, but combining this with the vertexcorrection (1.12), as we did before, the physical cross section isdσdΩphys=(dσdΩ)0exp[−αpiln(−q2m2)ln(−q2µ2)]exp[αpiln(−q2m2)ln(λ2µ2)]=(dσdΩ)0exp[−αpiln(−q2m2)ln(−q2λ2)].(1.23)This result contains information from all orders in perturbation theory, andagain only depends on physical parameters. This form can be shown toreproduce semiclassical predictions for the number of photons radiated as afunction of the parameters [26].This sort of cancellation seems to imply that in order to talk about physicaltransitions, we should speak only of processes including infinite amountsof soft radiation. In fact, with the above results, only these inclusive crosssections are nonzero and finite. That this radiation is inevitable was alreadydiscussed by Bloch and Nordsieck [30] long before the modern derivationsof the above results in e.g., [29, 31]. While our interpretation in terms ofunmeasurable soft modes is a practical one, formally it leaves much to bedesired. This discontent can be seen in the works of many authors whoattempt to redefine asymptotic states or the S-matrix in order to writedown a theory that never includes IR divergences in the first place [51–57].We note here that application of the functional formalism discussed next tothese sorts of “dressed state” constructions could be enlightening, and willbe the subject of future work.101.1. Infrared Problems in Quantum Field Theory1.1.2 Functional MethodsThe Bloch and Nordsieck approach came before the previous perturbativearguments, and involved understanding the behavior of an electron in abackground electric field. The background field was constrained only to in-volve low energy modes. This setup can be used in a more general functionalformalism originally due to Fradkin [2]. Here we will briefly discuss the ap-plication of this method to QED and show some of the peculiarities of itsresults. In particular, we derive the form of the electron propagator in thepresence of a slowly varying background field. This form of the propaga-tor can be used to derive an expression for the generating functional of thetheory, as is done in the next chapter for linearized quantum gravity. Ap-pendices C and D also discuss the relationship of the generating functionalto the S-matrix, which will be useful in deriving the soft theorems we havejust seen for gravitons. Our presentation closely follows that of Bogoliubovand Shirkov [36], though arguments can be found in other books and papersas well.1The equation of motion for the fermion field ψ in some fixed backgroundvector potential Aµ is[γµ (i∂µ + eAµ(x))−m]ψ(x) = 0. (1.24)In the Bloch-Nordsieck model, we approximate the gamma matrices (makingthis a scalar theory of QED) by making the replacementγµ → uµ, u2 = 1, (1.25)where uµ is a vector of constant numbers. Its interpretation in momentumspace is the velocity of the particle, uµ = pµ/m. This model leads to thedefinition of the electron propagator on the background[uµ (i∂µ + eAµ(x))−m]G(x, x′|A) = −δ(x− x′). (1.26)1See e.g. [34, 35] or the books of Popov [37] or Fried [1] and references therein.111.1. Infrared Problems in Quantum Field TheoryThis equation can be solved using the Schwinger/Fock “proper time” repre-sentation of the propagator, giving the formal solutionG(x, x′|A) = i∫ ∞0ds exp{is [uµ (i∂µ + eAµ(x))−m+ i] δ(x− x′)}.(1.27)The resulting form of G(x, x′|A) is discussed in detail when we do the calcu-lation for gravity. We then take the result, and functionally integrate overall possible long wavelength configurations of the background Aµ to get thefull electron propagator:G(x, x′) =∫DAeiS[A]G(x, x′|A), (1.28)where S[A] is just the action of the free vector field. The momentum spaceresult for the propagator in the long wavelength approximation turns out tobeG(p) ≈ 1m− u · p∣∣∣1− u · pm∣∣∣ζ , (1.29)where ζ ≡ − α2pi (3− ξ) +O(α2), and ξ is a gauge-fixing parameter. The formof ζ in the gauge where ξ = 0 was found by Solov’ev [38] to all orders in α,and this result was generalized to arbitrary gauges in [40]. This form showsa correction to the bare propagator by the multiplicative factor∣∣1− u·pm ∣∣ζ ,which can be expanded as∣∣∣1− u · pm∣∣∣ζ = 1− α2pi(3− ξ) ln∣∣∣1− u · pm∣∣∣+O(α2) (1.30)showing logarithmic corrections that are reminiscent of those we saw aftercancelling off IR divergences. The difference is that now we have none of theintuition of summing up indistinguishable processes to aid us in interpretingthis result. Looking closely, we see that the propagator no longer has asimple pole at u · p = m. This may appear troublesome, but in the gauge ofYennie and Fried [33], where ξ = 3, we see that this odd behavior disappearsentirely. Johnson and Zumino [32] remark that the multiplicative factor isdue only to the (gauge-dependent) description of scalar and longitudinal121.1. Infrared Problems in Quantum Field Theorymodes of Aµ, which should not be interpreted as physical degrees of freedom.Thus we should not be surprised that we can remove this factor withoutchanging the physics, as it is pure gauge. A proof of this at all ordersof perturbation theory was given by Braun [39], though that computationmissed a change in the wavefunction renormalization that was then correctedin [40].Before moving on, we mention that functional methods have also alreadybeen applied to the computation of eikonal amplitudes in QED [41, 42] andin gravity [45]. We will elaborate upon this approach later in this work, sowe do not discuss it in depth here. Such amplitudes result from summingan infinite number of Feynman diagrams [43]. In the two-fermion scatteringcase, these diagrams are the ladder type graphs obtained by ignoring vertexcorrections and the vacuum polarization of the exchanged photons. Theresult that we reproduce with the functional formalism for gravity is foundin [44]:iM = 8Ep∫d2x⊥e−iq⊥·x⊥(eiχ − 1) (1.31)where E is the energy and p is the center of mass momentum of both par-ticles, q is the momentum transfer, and the “eikonal” χ isχ ≈ −Gγ(s)Epln(µx⊥) (1.32)for γ(s) = 12[(s− 2m2)2 − 2m4], s is the usual Mandelstam variable, andµ is a graviton mass serving as an IR regulator. We refer to this paperafter recovering this result for further comments on bound state poles in theamplitude and relation to previous results.We will show in this work that the functional formalism is well-equippedto discuss all of the issues presented in this section. It contains all of thementioned IR divergences as well as their cancellation, and can at the sametime recover IR correlators and eikonal amplitudes. This thesis should beseen as a unification of these ideas and a first step in using the functionalapproach to better understand linearized quantum gravity.131.2. Theory of Quantized Gravity and Matter FieldsBut first - a quick review of linearized quantum gravity.1.2 Theory of Quantized Gravity and MatterFieldsShortly after the first major developments of quantum mechanics, an obviousquestion presented itself: how might quantum effects appear in Einstein’s(also still relatively new) generally covariant theory of gravity [8]? This ques-tion is at the heart of the field appropriately named quantum gravity.Carlo Rovelli’s “Notes for a brief history of quantum gravity” [12] helpfullybreak the myriad of approaches to the field into three categories, the “co-variant line of research,” the “canonical line of research,” and the “sum overhistories line of research.” The approach that we will focus on in this workfalls into the “covariant line of research,” in that it splits the spacetimemetric into a flat background, and a dynamical spin-two field that propa-gates on that background. In this line of research, the dynamical field isthen quantized, and interacts with quantum matter in a way described byconventional, flat spacetime quantum field theory. One simplifying aspect ofthis approach is that it straightforwardly avoids the question of how to gen-eralize quantum coordinate systems to allow a relativistic description. Wewill give a brief discussion of the subset of this line of work that is relevantto us, ignoring the other approaches except to refer the interested reader tothose related references in the Rovelli notes.To introduce the necessary elements of this sort of quantum theory of gravity,we first discuss the full and linearized Lagrangian derivation of the Einsteinfield equations (EFEs) in the presence of matter. We then discuss previ-ous attempts to understand how gravity can be quantized, and how eventhough the best we can do is a formally nonrenormalizable theory, we cangain understanding at low energies through the lens of effective field theory(EFT). This discussion includes a derivation of the Feynman rules for ourtheory. Throughout the discussion we will assume a basic understanding141.2. Theory of Quantized Gravity and Matter Fieldsof QFT, renormalization, and GR for the sake of brevity, and we assumestrictly unmodified Einstein GR taking the connection to be Levi-Cevitaand imposing only minimal coupling to matter.1.2.1 The Lagrangian Formulation of GRThe EFEs can be derived via the stationary action principle from the fol-lowing (Einstein-Hilbert) action:S[φ, gµν ] =∫d4x√−g[2κ2R+ Lφ]≡ SEH [gµν ] +∫d4x√−gLφ. (1.33)R is the Ricci scalar, obtained by fully contracting the Riemann curva-ture tensor. The Riemann tensor is built from the Levi-Cevita connectioncoefficients (Rµναβ ∼ ∂Γ − ∂Γ + ΓΓ − ΓΓ) which are in turn built from thespacetime metric gµν and its derivatives (Γ ∼ 12g[∂g+∂g−∂g]). κ2 = 32piG,g ≡ det(gµν), and Lφ is the Lagrangian density for any matter in the the-ory. Generically, φ can be shorthand for any types of matter fields, but forconcreteness and simplicity we take it to be a single real scalar field withLagrangian densityLφ[φ] = 12[gµν∇µφ∇νφ−m2φ2]. (1.34)The stationary action principle demands that δSEH/δgµν = 0, yielding im-mediately the EFEs:Rµν − 12gµνR =κ24Tµν (1.35)Tµν is the stress-energy tensor of the matter, defined byTµν =−2√−gδ(√−gLφ)δgµν. (1.36)This formalism leads to a consistent classical field theory, but how do wego about quantization? For our purposes, it will be sufficient to focus ontaking the limit of weak gravitational field.151.2. Theory of Quantized Gravity and Matter Fields1.2.2 LinearizationIf we focus our discussions on situations with only small spacetime curvature(e.g., a few elementary particles or mesoscopic systems) we can simplifythings a great deal. We will break the metric into two pieces: a constantMinkowski background and the dynamical field that encodes the departureof the full metric from that background. In other words,gµν(x) = ηµν + κhµν(x). (1.37)The factor of κ is for convenience in defining a canonically normalized quan-tum field theory.So far, this procedure is exact. At this point we introduce an approximationby assuming that |κhµν | 1. What we want is to reproduce the linearizedversion of the EFEs. To do this, we keep to leading order in κh in the puregravity action, and only the linear matter coupling term. After some tediousalgebra, the full action can be written as S[φ, hµν ] =∫d4xL, withL = Lg + Lφ + Lint, (1.38)Lg[h] ≡ 12∂µhαβ∂µhαβ − 12∂µh∂µh− ∂αhαγ∂βhβγ + ∂αhαβ∂βh (1.39)Lφ[φ] ≡ 12ηαβ∂αφ∂βφ− 12m2φ2 (1.40)Lint[φ, h] ≡ −κ2hµνTµν (1.41)Here, h ≡ ηµνhµν is the trace of the metric perturbation. For a scalar field,we haveTµν = ∂µφ∂νφ− ηµνLφ (1.42)Varying this new action w.r.t. hµν gives the linearized version of the EFEs.This sort of formalism is not new. The above manipulations are found inintroductory GR textbooks, and we refer to [10] for introduction and [9] forfurther information on the classical theory.161.2. Theory of Quantized Gravity and Matter Fields1.2.3 GravitonsTheories that quantize gravity around a flat background date back to at least1930 [12] with Rosenfeld [13, 14] offering a first attempt at applying quan-tization rules to the linearized EFEs. The sixties saw further development,notably due to the contributions of Feynman [15] and DeWitt [16–18]. Thearticle of Feynman provides a particularly readable account of the connec-tion of flat space quantization techniques to classical gravitational physicsvia computation of tree level scattering amplitudes. It also provides a senseof the first confusion resulting from the nonrenormalizability of quantizinggravity in the way we are about to. In order to understand these kinds of cal-culations, we will proceed with quantizing our linearized theory via the pathintegral. We can use path integral language to simply state the vacuum-to-vacuum amplitude for the graviton field in the presence of a classical sourceIµν . This amplitude is known as the generating functional (elaborated uponin the appendix). Writing this down for pure gravity, we haveZg[I] ≡∫Dh ei∫ Lg+i ∫ hµνIµν , (1.43)where h is now referred to as the graviton field. This path integral is Gaus-sian in h, but there is a problem. After integrating by parts, the pathintegral can be written in the formZg[I] ≡∫Dh e−i∫hµν∆−1µναβhαβ+i∫hµνIµν , (1.44)for a differential operator ∆−1, and to evaluate it we must find ∆. However,as is, the inverse of ∆−1 cannot be uniquely inverted. This is because hcontains redundant gauge degrees of freedom. Only once we pick a gaugecan we invert this operator.Under an infinitesimal change of coordinates, δxµ = −ξµ, the metric per-turbation is transformed according to δhµν = −∂µξν − ∂νξµ. The action isinvariant under this shift of the coordinate grid (gauge transformation), sowe are free to choose any ξ we would like. A convenient choice is the de171.2. Theory of Quantized Gravity and Matter FieldsDonder or harmonic gauge condition(hαβ),α =12ηαβh,α (1.45)which can be achieved with a choice of coordinates obeying∂2ξβ = (hαβ),α − 12ηαβh,α. (1.46)In this gauge, the generating functional can be evaluated exactly, but there isone further detail we must mention. In order to properly enforce this gaugecondition and stop ourselves from overcounting redundant gauge degrees offreedom in the path integral, we must first implement the Faddeev-Popovgauge fixing procedure. The result of this is [11]Zg[I] ≡∫DhDG[h]δ(Gβ(h))e−i∫hµν∆−1µναβhαβ+i∫hµνIµν , (1.47)where DG[h] is the Faddeev-Popov determinant corresponding to the gauge-fixing function Gβ(h), and the delta function simply enforces the constraint(1.45), meaning that for the de Donder gauge we haveGβ(h) = (hαβ),α − 12ηαβh,α. (1.48)For linearized gravity, the determinant only effects a change of overall nor-malization, so we henceforth ignore it. This leavesZg[I] ≡∫Dh δ(Gβ(h))e−i∫hµν∆−1µναβhαβ+i∫hµνIµν , (1.49)which adequately defines the inverse of ∆−1 and allows the integral to beperformed, givingZg[I] = exp{i2∫d4x∫d4y Iαβ(x)∆αβσρ(x− y)Iσρ(y)}, (1.50)where the free graviton propagator ∆ is the Green function of the linearizedEFEs and is given by181.2. Theory of Quantized Gravity and Matter Fields∆αβσρ(x− x′) ≡ x x′ =∑keik·(x−x′)k2Pαβσρ, (1.51)where Pαβσρ ≡ 12 [ηασηβρ + ηαρηβσ − ηαβησρ], and we now have an effectivequantum field theory for weak-field gravity on its own. The theory of free(i.e., no coupling to gravity) scalar matter is even simpler, withZφ[J ] ≡∫Dφ ei∫ Lφ+i ∫ φJ=∫Dφ e−i∫φG−10 φ+i∫φJ ,(1.52)Where Lφ is given by equation (1.40), and we are now in Minkowski space.The differential operator in the exponent is easy to write down,G−10 = ∂2 +m2, (1.53)as is its inverse,G0(x− x′) ≡ x x′ =∑keik·(x−x′)k2 −m2 . (1.54)G0 is the free scalar propagator, and we are almost done constructing theFeynman rules for this theory.From the interaction part of the Lagrangian density, Lint, we can imme-diately write down the vertex (figure 1.3), here in momentum space, withq = p1 − p2:τµν(p1, p2; q) = κ(p1(µp2ν) −12ηµν [p1 · p2 −m2]) (1.55)191.2. Theory of Quantized Gravity and Matter Fieldsp1 p2qµνFigure 1.3: Graviton-matter vertex.So we have established the diagrammar for this theory. This allows us toproceed and calculate gravitational scattering amplitudes using Feynmandiagrams [24]. When we do so, we run into the usual UV divergences fromcomputing loop integrals. These can be absorbed into renormalized param-eters which are compared to experiment in order to measure their physicalvalues. This makes it so physical quantities can still be predicted in spite ofthe infinities spit out by the diagrammatics. In order for a theory to havepredictive power, it is naively required that the number of necessary renor-malized constants needed to remove UV divergences to all orders is finite.Unfortunately, this requirement is not satisfied by gravity [19]. To eliminateUV divergences at every order would require an ever-increasing number ofcounterterms. The naive argument then says that the theory should have nopredictive power. Can we ever hope to make predictions with such a theory,or is this line of attack dead?2Thankfully, it seems like we can recover predictive power if we think of lin-earized gravity as an effective field theory [11, 20, 21]. An EFT is one thatis understood to be the low energy limit of some (perhaps unknown) micro-scopic theory that is valid at all length scales. From this perspective, thoughrenormalization forces upon our theory infinitely many higher order interac-tion terms in the Lagrangian (including terms beyond the Einstein-Hilbertterm), they are all suppressed by ever-increasing powers of κE ∼ E/MP ,where E characterizes the energy scale at which we are interested in apply-ing our theory and MP ∼ 1018GeV is the Planck mass. This means that,although we are formally in trouble, we can truncate our theory as we havedone if we promise to only apply it at energy scales well below the Planck2Rovelli says it is dead. Specifically, he claims it died in 1975 [12].201.2. Theory of Quantized Gravity and Matter Fieldsscale. The EFT perspective also has the pleasing consequence that anyUV complete theory of quantum gravity should make predictions that agreewith ones computed with our truncated theory in the limit of low energy.This method of reinterpretation gets concrete results. These calculationshave been shown to reproduce the classical Newtonian potential in two-bodyscattering processes, and quantum corrections to the Newtonian potentialhave been derived.3 This should support our intuition that the theory isvalid despite its infinitely many problem terms at higher orders.So we have discovered a low energy effective theory that reproduces the New-tonian potential, and contains quantum corrections. We are finally ready toinvestigate the same problems that we saw arise in QED in the last section.To that end we will proceed to apply the same functional techniques we sawthere to our quantum theory of gravity.3For more, see [22], and also [23] and references therein.21Chapter 2Eikonal Methods forLinearized GravityIn this section we expand upon the introduction to functional methods inQED, as we apply the technique to linearized quantum gravity. Our pre-sentation is similar to those of Fradkin and Fried [1, 2]. We will derivethe IR form of the scalar propagator with soft gravitons functionally inte-grated out, and explain the details of the functional method along the way.Another attractive aspect of the functional approach we have yet to fullydiscuss is that it also allows a derivation of an incredibly convenient form ofthe generating functional of the theory in the eikonal limit. This facilitatescomputation of not just the infrared effective propagator, but in principleall of the n-point functions of the theory. This too is done in this chapter.We will not really need an expression for the generating functional in thislimit to derive many results in this work, but we write it down anyway, andshow how to extract any desired correlators. Why? As Schwartz [5] puts it,“The generating functional is the holy grail of any particular field theory:if you have an exact closed-form expression for Z for a particular theory,you have solved it completely.” Our form is close to this ideal - while notexact, it is justified in the soft limit, and while not closed-form in its finalincarnation, still very simple to use. We hope that this form for Z can beof use in other applications beyond the scope of this work.Our implementation of the functional method involves two explicit assump-tions. The first is that at any point, as matter moves through spacetime, thebackground graviton field is slowly varying compared to the dynamics of the222.1. Equation of Motion for the Scalar Fieldparticle. This assumption is certainly well motivated for non-relativistic pro-cesses, and also for the isolation of soft graviton effects, which are necessarilylong wavelength. The second assumption is that at any point, the likelihoodthat a virtual graviton will pair produce scalar particles is negligible. Inother words, we ignore all loop corrections to the graviton propagator, andsimply use the bare one everywhere. This too is justified by the focusing ofour attention to soft effects. Virtual quanta would need to attain energiesof order 2mc2 as in figure 2.1 in order for us to include diagrams involvingscalar loops, but this energy scale is far above that which we defined assoft.qmqFigure 2.1: In order for virtual gravitons to be modified by loop correctionslike this one, where a pair of massive scalar particles are produced, themomentum q must be of order 2mc2.With our assumptions stated, we begin by asking the simplest question ofall: how does the scalar field behave in the presence of some particulargraviton background?2.1 Equation of Motion for the Scalar FieldWith the effective Lagrangian density L[φ, h] = Lg[h]+Lφ[φ]+Lint[φ, h], wecan isolate the behavior of the matter, conditioned on a particular configu-ration of the metric. What we will do is “freeze” the perturbation field h insome arbitrary configuration, and get an equation of motion for φ. Varyingthe action w.r.t. a small variation in the field φ → φ + δφ and demanding232.2. The Generating Functionalthat δS = 0 gives an equation of motion for φ:{∂2 +m2 + κhαβ(x)Kˆαβ}φ(x) = 0, (2.1)Kˆαβ ≡ −∂α∂β + 12ηαβ(∂2 +m2). (2.2)where again we have chosen to enforce the de Donder gauge condition. Thenthe propagator (Green function) for a scalar on a fixed graviton backgroundis defined by: {G−10 + κhαβ(x)Kˆαβ}G(x, x′|h) = −δ4(x− x′) (2.3)We will need this in what follows and its eikonal form will be discussed later.For now we proceed to show how we can derive the generating functional ofthe theory in terms of this quantity.2.2 The Generating FunctionalIf J(x) and Iαβ(x) are again arbitrary classical sources of the scalar field φ(x)and the graviton field hαβ(x) respectively, the full generating functional forthe theory with the Lagrangian given above isZ[J, I] =∫Dφ∫Dh eiSφ+iSg+iSint+i∫φJ+i∫hI . (2.4)Recall, Sint[φ, h] = −κ2∫d4xhαβTαβ. After integrating by parts, this canbe rewritten asSint[φ, h] = −κ2∫d4xhαβφKˆαβφ. (2.5)Now in this form, we write Z asZ = eiSint[ δδJ , δδI ]∫Dφ∫DheiSφ+iSg+i∫φJ+i∫hI , (2.6)242.2. The Generating Functionalwhere we have made the substitutions φ(x) → −iδδJ(x) and hαβ(x) → −iδδIαβ(x) .Written this way, the path integrals decouple and each can be evaluatedexplicitly, as done in the previous chapter. This givesZ = eiSint[ δδJ , δδI ]Zφ[J ]Zg[I]. (2.7)Now, notice that eiSint is simply a linear shift operator in I, and a quadraticshift operator in J . Because the free generating functionals factorize and areGaussian, either of these operators can be applied directly. The linear shiftin I is easier, but first evaluating the quadratic shift in J is more helpful.Applying this givesZ = exp{−12∫d4x ln(1 + κG0(x, x)Kˆαβ−iδδIαβ(x))}×exp i2∫d4x∫d4yJ(x) G0(x, y)1 + κG0Kˆαβ−iδδIαβ J(y)Z0φ[J ]Z0g [I].(2.8)One then recognizes that the quantity G0(x,y)1+κG0Kˆαβ−iδδIαβis the (symbolic) solu-tion to the equation that defines the propagator G(x, y|h) on a fixed back-ground perturbation field, with the field hµν(x) replaced by −iδδIµν(x) . Thismeans that Z can be written asZ = exp{i2∫d4x∫ κ0dg G(x, x|g δδI)KˆαβδδIαβ(x)}× exp{i2∫d4x∫d4y J(x)G(x, y| δδI)J(y)}Z0φ[J ]Z0g [I].(2.9)The first exponential “interaction operator” above describes the polariza-tion of gravitons ([1], ch.3) which here means diagrams with scalar loops(remember, we have removed all the nonlinearity at the level of the action,so there are no graviton-only polarization contributions). Because of this,we ignore that term in the calculations that follow, setting it to 1, as phys-252.3. Propagator on a Fixed Backgroundical arguments presented at the beginning of this chapter say there shouldbe no such diagrams in the infrared.Now all that we need is an appropriate expression for the scalar propagatorin a fixed, slowly varying background field.2.3 Propagator on a Fixed BackgroundRemember, the propagator we want to find is defined by{G−10 + κhαβ(x)Kˆαβ}G(x, x′|h) = −δ4(x− x′). (2.10)In momentum space, with G(x, x′|h) = ∑k eik(x−x′)Gk(x|h), the propagatorobeys {G−10 (k)− Uˆ}Gk(x|h) = 1. (2.11)Here we are expanding about the bare propagator, G0(k) =1k2−m2 , andexplicitly:Uˆ ≡ ∂2 + 2ikµ∂µ+κhαβ[kαkβ − 2ikα∂β − ∂α∂β − 12ηαβ(k2 −m2 − ∂2 − 2ikµ∂µ)] (2.12)At this point, it is also useful to scale the above relation by the scalarmass: {mG−10 (k)−mUˆ}Gk(x|h) = m (2.13)The reason for this is made clear below. Using the Schwinger/Fock “propertime” representation, we can write the bare propagator asG0(k) = im∫ ∞0dse−ism(k2−m2) ≡ im∫ ∞0dsG0(k, s), (2.14)where the exponent is to be given a small negative imaginary part, and it is262.3. Propagator on a Fixed Backgrounduseful to note thati∂sG0(k, s) = m(k2 −m2)G0(k, s) = mG−10 (k)G0(k, s). (2.15)It is also convenient to express the full propagator asGk(x|h) = im∫ ∞0dsG0(k, s)Y(k, s, x|h), (2.16)such that Y acts to weight the free propagator term under the proper timeintegral. In order to do this, we must be able to satisfy{mG−10 (k)−mUˆ}i∫ ∞0dsG0(k, s)Y(k, s, x|h) = 1, (2.17)and it turns out we can, if the weighting factor obeys the Schro¨dinger equa-tion− i∂sY = mUˆY, Y(s = 0) = 1. (2.18)In deducing the form of Y, we use the ansatzY ≡ eχ, (2.19)with χ inheriting all of the dependencies of Y.Now, for this scalar theory, the equation of motion for χ is nonlinear andgenerally intractable. Progress can be made, however, by expressing χ as apower series. In order to do this, note that the coupling κ =√32piG can beexpressed in natural units as κ =√32pi/MP , with MP the Planck mass. Thescaling of the perturbation Uˆ by the scalar mass m gives naturally a smalldimensionless parameter mκ =√32pi(m/MP ) 1. So to isolate the eikonalbehavior of the propagator, expand χ as a power series in the dimensionless272.3. Propagator on a Fixed Background“coupling” in the spirit of the WKB technique:χ ≡∞∑n=1(mκ)nχn (2.20)To first order in mκ, this gives an equation of motion for χ1:− i∂sχ1 = m[∂2 + 2ikµ∂µ]χ1 + hαβ(x)[kαkβ − 12ηαβ(k2 −m2)] (2.21)Higher order terms can be found in a similar manner. One interesting thingto note is that, unlike in a linear theory (e.g. QED), having to perform thisexpansion at this stage due to the second-order equation of motion couldlead to a different source of subleading soft effects. However this has not yetbeen investigated. The ∂2 term can also be dropped at this stage due to theassumption of a slowly-varying background field.4 We then write χ in termsof its Fourier transform, χ1 =∑q eiq·xχ˜1(q), in which case the solution forχ˜1(q) readsmκχ˜1(q) = im∫ s0ds′e−2is′m(k·q)h˜µν(q)τµν(k), (2.22)where the graviton-scalar vertex in de Donder gauge is again τµν(p1, p2; q) =κ(p1(µp2ν) − 12ηµν [p1 · p2 −m2]). The appearance of τµν(k) ≡ τµν(k, k; 0) isa clear manifestation of the eikonal physics here. Imposing momentum con-servation at the vertex and setting p1 = p2 = k implies that, if one is totruly take this object as a vertex, the momentum of the emitted gravitonis identically zero. This can also be thought of as a manifestation of theinconsistency of the linearized theory, in which the gravitons do not sourcethemselves (i.e. one incorrectly assumes that the stress-energy of gravitonsis zero).4If we do not drop this term here, we get a slightly more complicated form factor later.This would consist of replacing the factor∫ s0ds′e−2is′m(k·q) with∫ s0ds′e−is′m(q2+2k·q). Thisdoes not change the final results of our work, so for our purposes these forms are equivalent.However if one wishes to consider slightly larger graviton momenta, this difference becomesmore important.282.4. Applying the Interaction OperatorsFinally, keeping just the first order eikonal correction, we have thatGk(x|h) ≈ im∫ ∞0ds e−ism(k2−m2)+mκ∑q eiq·xχ˜1(q), (2.23)and the explicit dependence on the mass scale can be seen to be trivial bychoosing the integration variable to be sm rather than s (and similarly fors′ in the form factor of χ˜1), to getGk(x|h) ≈ i∫ ∞0ds e−is(k2−m2)+κ∑q eiq·xχ˜1(q). (2.24)How is this different from perturbation theory? In our ansatz (2.19),we chose to leave the s-dependence of χ arbitrary. Choosing a form likeY ≡ e−isχ¯ (2.25)where χ¯ is no longer a function of proper time, would giveG = 1k2 −m2 + χ¯ . (2.26)We would find that, to first order, χ¯ is just the one-loop self energy of thescalar. So in this way, we could immediately recover the usual perturba-tion theory by keeping ever-increasing orders of corrections to χ¯. However,simply relaxing the dependence on proper time gives strictly nonperturba-tive results and we now proceed to investigate what effect this has on thegenerating functional and correlators of the theory.2.4 Applying the Interaction OperatorsSo, then, the propagator given on a fixed background, to first order in mκ,has the formGk(x|h) ≈ i∫ ∞0ds e−is(k2−m2)−∑q eiq·xf(q)τµν(k,k)h˜µν(q), (2.27)292.4. Applying the Interaction Operatorswhere the form factor isf(q) ≡ −i∫ s0ds′e−2is′(k·q) =e−2is(k·q) − 12k · q . (2.28)Plugging (2.27) into (2.9) and setting the first exponential to one, we seethat the generating functional in the infrared regime has the approximateformZ ≈ exp{i2∫d4x∑keik·xJ(x)J˜(k)Gk(x| δδI)}× exp{i2∑qI˜µν(q)∆µναβ(q2)I˜αβ(−q)},(2.29)or in convenient shorthand,Z ≈ e i2∫JG(|−iδδI)Jei2∫I∆I . (2.30)This gets a little messy, but since the propagator here is restricted to terms inthe exponent that are linear in κh, the result is a straightforward applicationof the properties of linear shift operators. To see this, write the aboveas∞∑n=0(−1)nn![12∫d4x∑keik·xJ(x)J˜(k)×∫ ∞0ds e−is(k2−m2)+i∑q eiq·xf(q)τµν(k) δδI˜µν (q)]nZ0g [I˜µν(q)](2.31)=∞∑n=0(−1)nn!n∏a=1[12∫d4xa∑kaeika·xaJ(xa)J˜(ka)∫ ∞0dsa e−isa(k2a−m2)]× ei∑q∑na=1 eiqa·xafa(q)τµν(ka) δδI˜µν (q)Z0g [I˜µν(q)].(2.32)302.5. CorrelatorsRecognizing this as a shift operator -exp{i∑qeiq·xf(q)τµν(k)δδI˜µν(q)}Z0g [I˜µν(q)]= Z0g[I˜µν(q) + ieiq·xf(q)τµν(k)] (2.33)- and writing out the shifted Z0g gives the explicit form for the generatingfunctional:Z =∞∑n=0(−1)nn!n∏a=112∫d4xa∑kaeika·xaJ(xa)J˜(ka)∫ ∞0dsa e−isa(k2a−m2)Z0g [I]× exp{−n∑a=1∑qe−iq·xafa(−q)τµν(ka)∆µναβ(q2)I˜αβ(q)− i2n∑a,b=1∑qeiq·(xa−xb)fa(q)fb(−q)τµν(ka)∆µναβ(q2)ταβ(kb)}(2.34)This is not quite the “exact, closed-form expression” we may dream of, butin fact the form (2.30) is practically just as useful. To demonstrate this, wenow run through a couple examples of finding n-point correlators.2.5 CorrelatorsNow that we have an explicit form for Z in the soft graviton regime, we canuse it to find the IR form of whichever correlators we are interested in. Inparticular, we focus here on finding the first couple of correlators, G2 andG4. DefiningZ[J, I] ≡ eiW[J,I], W = −ilnZ, (2.35)the correlators for the scalar field areGn(x1, . . . , xn) = (−i)nδnZ[J, I]δJ(x1) . . . δJ(xn)|I=J=0. (2.36)312.5. Correlatorsand the connected correlators areG(c)n (x1, . . . , xn) =(−i)nδniW[J, I]δJ(x1) . . . δJ(xn)|I=J=0. (2.37)In practice, it is most convenient to use the real space form of (2.30) tocalculate these. Doing things this way results in expressions that are simplyrepeated applications of linear shift operators, so all of the n-point functionsfor φ are simple to recover.2.5.1 The Two-Point FunctionFollowing (2.30),−iδZδJ(x)=[−∫d4a J(a)G(x, a| δδI)]Z. (2.38)Rinsing and repeating:(−i)2δ2ZδJ(x)δJ(x′)=[−iG(x, x′| δδI) + 0]Z, (2.39)where the bold 0 indicates more terms that vanish when the auxiliary sourcesare set to zero after applying all functional derivatives. Now all that is left isto evaluate G acting on Z, as done before in (2.33), and turn off the sources.The result isG2(r) =∑keik·r∫ ∞0ds e−is(k2−m2)× exp{− i2∑qf(q)f(−q)ταβ(k)∆αβµν(q2)τµν(k)},(2.40)where r stands for the difference x− x′. Though the propagator found witha fixed background was not translationally invariant, this correlator is afterintegrating out background configurations.322.5. Correlators2.5.2 The Four-Point FunctionComputing the two particle correlator is more messy, but just as straight-forward. By definition,G4(w, x, y, z) = (−i)4δ4ZδJ(w)δJ(x)δJ(y)δJ(z)|I=J=0, (2.41)and after computing this we getG4(w, x, y, z) = G(w, x|y, z) +G(w, y|x, z) +G(w, z|x, y), (2.42)whereG(w, x|y, z) ≡∑k1∑k2eik1·(w−x)eik2·(y−z)∫ ∞0ds1∫ ∞0ds2 e−is1(k21−m2)e−is2(k22−m2)×exp{− i2∑qf1(q)f1(−q)ταβ(k1)∆αβµν(q2)τµν(k1)}××exp{− i2∑qf2(q)f2(−q)ταβ(k2)∆αβµν(q2)τµν(k2)}××exp{−i∑qeiq·(w−y)f1(q)f2(−q)ταβ(k1)∆αβµν(q2)τµν(k2)}.(2.43)Any correlator can be computed in this manner, and using this first approx-imation to Z makes each of them straightforward.332.6. Infrared Behavior of the Two-Point Function2.6 Infrared Behavior of the Two-PointFunctionIn addition to pulling G2 from Z in the manner above, one can also recover itby integrating out the fluctuations of the metric perturbation directlyG(x− y) =∫Dhµν P[h]G(x, y|h), (2.44)in analogy with ordinary probability theory. P is a probability amplitudefor a given configuration of h. By the definition of the full correlator for φ,P is found to be P[h] = eiSg [h], with Sg[h] the pure gravity action, whichin this treatment is only quadratic in h. The integral is then Gaussian in hand is straightforward. From either derivation, we get the formG(k2) =∫ ∞0ds e−is(k2−m2)exp{− i2∑qf(q)f(−q)ταβ(k)∆αβµν(q2)τµν(k)}.(2.45)Evaluating the integral in the phase - seen in Appendix B - and usingταβPαβµντµν ≈ κ24 k4 ≈ κ24 m4 near the mass shell,G(k2) ≈∫ ∞0ds e−is(k2−m2)− κ232pi2m2[isΛ−ln(sΛ)]. (2.46)In the new phase, the term linear in s should be thought of as a mass renor-malization. However at this point it must be remarked that these results aregauge-dependent, and as discussed in [1], it should simply be assumed thatall physical mass renormalization has been done at this stage, and that thosefull results should rightfully be gauge-independent. Additionally, writingln(sΛ) = ln(sm) + ln(Λ/m) gives a divergent wavefunction renormalization.Finally, the renormalized propagator looks likeG(k2) =∫ ∞0ds e−is(k2−m2)+ κ232pi2m2 ln(sm) =∫ ∞0ds e−is(k2−m2)(sm)ζ ,(2.47)342.6. Infrared Behavior of the Two-Point Functionwhere ζ ≡ Gm2pi , and thus in harmonic gauge:G(k2) = −iΓ(1 + ζ)(−im)ζ [k2 −m2]−(1+ζ) (2.48)We emphasize that this result as presented is valid only in harmonic gauge.We have suppressed the dependence on a gauge-fixing parameter to simplifythe index gymnastics. However as in the scalar QED result in the introduc-tion, it is still possible to reduce this propagator to a form with a simple poleby a suitable gauge choice, discussed in [66, 67]. In this way one gets rid ofspurious apparent IR divergences in the propagator. It would be interestingto repeat the calculations performed in this work using this divergence-freegauge.The key results of this chapter are equations (2.30), (2.45), and (2.48). Thefirst gives an eikonal representation of the generating functional, and answersthe general question of how the matter-gravity theory behaves in the lowenergy limit. The last gives a nonperturbative result for the new effectivepropagator of the scalar field which has the same basic form as the result forQED. It does this through the help of the convenient form (2.45). Thougha similar form for the QED propagator is known, we have presented a newapproach to calculating (2.48), which matches the known result, e.g., eq.4.7 of [66].The form (2.45) is crucial because it allows for efficient computation of morephysical quantities, like S-matrix elements. This connection is what will al-low us to make contact with experiment (via probabilities from squaredS-matrix elements) and to current theoretical discussion involving soft the-orems. To that end, we proceed by discussing how to use this functionalformalism to compute scattering amplitudes. This will also be a convincingproof that this formalism encodes the same soft physics as diagrammaticapproaches.35Chapter 3Amplitudes and SoftTheoremsSo far we have obtained expressions for the generating functional and cor-relators for our theory. How can we use this technology to derive S-matrixelements? In this section we will show how to compute these amplitudes,and how the soft graviton theorem comes about naturally from these calcula-tions. In particular, we show how it emerges as a consequence of the eikonalformulation of the theories in the specific cases of bremsstrahlung facilitatedby a classical potential and two-body scattering. We make some commentsabout the cancellation of IR divergences in these cases, but those results aremore or less the same as those discussed for QED in the introduction.The important thing about the soft photon results at the beginning of thiswork one should note for this chapter is summarized in equation (1.21). Theeffect of adding a soft photon emission to an amplitude is a divergent multi-plicative factor. The same is true for gravity, with only slight modificationto the resulting factor [29]. The soft graviton correction takes some knownamplitude M toM×√8piG2(2pi)3/2√2|q| m∑i∗µν(p′i)µ(p′i)νp′i · q − iδ−n∑j∗µν(pj)µ(pj)νpj · q + iδ ≡M× κΩ.(3.1)363.1. Gravitational Bremsstrahlung3.1 Gravitational BremsstrahlungHere we will give the simplest possible example of gravitational bremsstrahlung- a massive scalar particle in a small external metric perturbation field hextemitting n (soft) gravitons, with momenta {qn} and polarizations labeledby {λn}.3.1.1 Constructing the AmplitudeThe amplitude for this process (using B for Bremsstrahlung) isBn = IN〈p′; q1, λ1; · · · ; qn, λn|p〉IN = OUT〈p′; q1, λ1; · · · ; qn, λn|S|p〉IN , (3.2)and we will drop the “IN/OUT” subscripts from here on out. First, considerthe case of the emission of a single graviton. The generalization to multiplegraviton emission will be easy. We construct the states by〈p′; q, λ| = 〈0| aˆ(p′)bˆ(q) (3.3)and|p〉 = aˆ†(p) |0〉 . (3.4)The a creation/annihilation operators create/annihilate scalar particles, whilethe b operators are for the gravitons. We also have (see the appendix)thatS = : e∫φINDˆ δδJ+∫hµνIN Kˆ δδIµν : Z[J, I]|J=I=0 ≡ S¯Z|0, (3.5)where the : (· · · ) : denotes normal ordering of operators, and Z is the fullgenerating functional. Dˆ and Kˆ are the (free) inverse propagators for thescalar field and the graviton field respectively (to be identified with thedifferential operators previously called G−10 and ∆−1), and the IN fieldsareφIN(x) =(3)∑k1√2ωp[aˆ(k)e−ik·x + aˆ†(k)eik·x]; ω2p = k2 −m2, (3.6)373.1. Gravitational BremsstrahlungandhµνIN(x) =(3)∑k1√2|~k|∑λ[µν(k, λ)bˆ(k)e−ik·x + µν∗(k, λ)bˆ†(k)eik·x]. (3.7)Here the µν = µν are the graviton polarization vectors. In this construc-tion, the amplitude isB1 = 〈0| aˆ(p′)bˆ(q, λ) (S¯Z|0) aˆ†(p) |0〉 . (3.8)Commuting the creation and annihilation operators through the S-matrixgives for B1B1 = 〈0|(∫dx[bˆ(q, λ), hµνIN(x)]Kˆ δδIµν(x))(∫dy[aˆ(p′), φIN(y)] Dˆ δδJ(y))×(∫dz[φIN(z), aˆ†(p)]Dˆ δδJ(z))(S¯Z|0) |0〉(3.9)= (2pi)−9/2∗(q, λ)√2|~q|12√ωpωp′∫dx eiq·xKˆx∫dy eip′·yDˆy∫dz e−ip·zDˆz× δδI(x)δδJ(y)δδJ(z)Z[J, I]|0.(3.10)3.1.2 Functional Eikonal LimitTo proceed, we must employ some particular expression for the generatingfunctional. Let us use the eikonal form that we have already derived, eq.(2.30):Z[J, I] ≈ e i2∫JG(|−iδδI)Jei2∫I∆I , (3.11)where we had usedG(x, y|h) ≈∑keik·(x−y)i∫ ∞0ds e−is(k2−m2)+ i2τµν(k)∫ s0 ds′ hµν(y+s′k), (3.12)383.1. Gravitational Bremsstrahlungwith τµν(k) ≈ κkµkν (as the scalar is almost on shell: k2−m2 ≈ 0). Lookingat (3.10), we see that the two functional J-derivatives bring down one G,and we replace the h-dependence of G by −iδδI , so that G ∼ e−i∫f δδI . G thenis applied to ei2∫I∆I . After using the functional identity (A.5), we getB1 = i(2pi)−9/2 ∗(q, λ)√2|~q|12√ωpωp′[∫dx eiq·xδδh(x)e−i2∫δδh∆ δδh×∫dy eip′·yDˆy∫dz e−ip·zDˆz]G(y, z|h+ hext)|h=0.(3.13)Note that we have finally made the dependence on hext explicit in theabove.3.1.3 On-Shell ResultsAssuming that |κhext| is small, we keep the first perturbative contributionof the classical source by making the replacementG(y, z|h+ hext)→ i∫duG(y, u|h)τ · hext(u)G(u, z|h). (3.14)This leaves two mass-shell amputations to be performed, with the results∫dy eip′·yDˆyG(y, u|h)|p′2→m2 = eip′·uei2∫∞0 ds τ′·h(u−sp′),∫dz e−ip·zDˆzG(u, z|h)|p2→m2 = e−ip·uei2∫∞0 ds τ ·h(u+sp),(3.15)where we have assumed that the scalar follows straight line paths from y tox, defining the eikonal limit. The seemingly complicated eikonal correlatorshave thus given simple Wilson-line-type factors that will be acted upon bythe operators corresponding to real graviton radiation. The amplitude then393.1. Gravitational BremsstrahlungbecomesB1 = i(2pi)−9/2 ∗(q, λ)√2|~q|12√ωpωp′[∫du eiu·(p′−p) τ · hext(u)×∫dx eiq·xδδh(x)e−i2∫δδh∆ δδh]ei∫fh|h=0,(3.16)wherefµν(w, u) ≡ 12∑qe−iq·(w−u)∫ ∞0ds[τµνeisp·q + τ ′µνe−isp′·q]≡ f inµν(w, u) + foutµν (w, u).(3.17)Now application of the quadratic shift e−i2∫δδh∆ δδh is trivial, givingB1 = i(2pi)−3 12√ωpωp′∫du eiu·(p′−p) τ · hext(u)e i2∫f∆f×((2pi)−3/2∗(q, λ)√2|~q|∫dx eiq·xδδh(x)ei∫fh|h=0).(3.18)Before expanding upon this result, let us investigate what happens when ngravitons are emitted.3.1.4 Multiple Graviton EmissionGeneralizing to n-graviton radiation, the amplitude becomes simplyBn =i(2pi)−3 12√ωpωp′∫du eiu·(p′−p) τ · hext(u)e i2∫f∆f×n∏m((2pi)−3/2∗(qm, λm)√2| ~qm|∫dxm eiqm·xm δδh(xm)ei∫fh|h=0).(3.19)403.1. Gravitational BremsstrahlungFor each graviton radiated, as q → 0, we get∫dx eiq·xδδhµν(x)ei∫fh|h=0 =√8piG[p′µp′νp′ · q − iδ −pµpνp · q + iδ]+O(q0)(3.20)so that the full amplitude becomesBn =(i(2pi)−312√ωpωp′[τ · h˜ext(p− p′)]ei2∫f∆f)×n∏m( √8piG(2pi)3/2√2| ~qm|[µν∗m p′µp′νp′ · q − iδ −µν∗m pµpνp · qm + iδ]).(3.21)In other words,Bn = Beik.0 ×n∏m( √8piG(2pi)3/2√2| ~qm|[µν∗m p′µp′νp′ · q − iδ −µν∗m pµpνp · qm + iδ]). (3.22)We have just derived an example of the soft graviton theorem.3.1.5 Virtual Graviton ExchangeThe rest of the amplitude Beik.0 also merits some comments. We haveei2∫f∆f = ei2∫[f in+fout]∆[f in+fout]. (3.23)The terms generated byei2∫f in∆f in , ei2∫fout∆fout (3.24)correspond to a divergent wavefunction renormalization, as well as a renor-malization of the mass. To see this, write out the integrals in each of theseexponentials and note that they are the same for each particle individu-ally as the ones encountered in the previous chapter when deriving the IRform of the propagator (see the discussion just before equation (2.47)). Inthis case however, we have already done the overall proper time integrals413.2. Two-Body Scatteringwhen performing the mass shell amputation, leaving no dependence on s,and these renormalization constants factor out. As such, we simply assumerenormalization has been carried out and drop these parts. The rest,ei∫f in∆fout , (3.25)generates vertex corrections at the perturbative point of contact with theexternal field. Namely, the eikonal limit implicitly sums all graphs like thefollowing:Beik.0 =pp′= + + + + · · ·(3.26)Though this form contains divergences, these divergences cancel with thosecoming from real emission. To see this, simply evaluate the integral inei∫f in∆fout and notice that it corresponds to the virtual infrared divergencescomputed in section 3 of [29], and is precisely the gravitational analog ofthe Sudakov calculation [27]. With our result for the soft factors in equa-tion (3.22), the calculation of real divergences in section 4 of [29] and thesubsequent cancellation of these divergences proceeds identically to those in[29], section 5.3.2 Two-Body Scattering3.2.1 Constructing the Eikonal AmplitudeHere we will be concerned with the radiation of soft gravitons alongside a2 → 2 scattering process. This amplitude (denoted A to distinguish fromthe previous discussion) is given byAn = 〈p′1; p′2; q1, λ1; · · · ; qn, λn|p1; p2〉 , (3.27)423.2. Two-Body Scatteringand for convenience we start with the single-emission case:A1 ≡qp′2p′1p1 p2(3.28)This amplitude has the expressionA1 = (2pi)−15/2 ∗(q, λ)√2|~q|14√ω1ω′1ω2ω′2∫dx eiq·xKˆx×∫dy1 eip′1·y1Dˆy1∫dy2 eip′2·y2Dˆy2∫dz1 e−ip1·z1Dˆz1∫dz2 e−ip2·z2Dˆz2× δδI(x)δδJ(y1)δδJ(y2)δδJ(z1)δδJ(z2)Z[J, I]|0.(3.29)Using the same manipulations as we did for B, we can rewrite this asA1 = (2pi)−15/2 ∗(q, λ)√2|~q|14√ω1ω′1ω2ω′2∫dx eiq·xδδh(x)e−i2∫δδh∆ δδh×∫dy1 eip′1·y1Dˆy1∫dy2 eip′2·y2Dˆy2∫dz1 e−ip1·z1Dˆz1∫dz2 e−ip2·z2Dˆz2× [G(y1, z1|h)G(y2, z2|h) + (y1 ↔ y2)]|h→0.(3.30)We will from now suppress the coordinate dependence by G(y1, z1|h) ≡ G1,etc. Now, again following [1], ch. 10, we compute not A, but ∂A/∂κ2. Thisis so that we can again produce similar Wilson line factors that generate thesoft radiation dependence of the amplitude. We also remind ourselves that433.2. Two-Body Scatteringin this notation, τ ∼ κ. After some work,∂A1∂κ2=(2pi)−15/2∗(q, λ)√2|~q|14√ω1ω′1ω2ω′2∫du1∫du2 ∆αβσρ(u1 − u2)∫dx eiq·xδδh(x)e−i2∫δδh∆ δδh×∫dy1 eip′1·y1Dˆy1∫dy2 eip′2·y2Dˆy2∫dz1 e−ip1·z1Dˆz1∫dz2 e−ip2·z2Dˆz2× (G1τ(1)αβκG1)(G2 τ(2)σρκG2)|h→0 + 0.(3.31)The 0 indicates terms that do not vanish identically, but will not contributeto the final results at leading order.3.2.2 On-Shell ResultsUsing (3.15), we put the scalar particles on shell, giving∂A1∂κ2=(2pi)−614√ω1ω′1ω2ω′2∫du1∫du2 ∆αβσρ(u1 − u2)τ(1)αβ τ(2)σρκ2eiu1·(p′1−p1)eiu2·(p′2−p2)× e i2∫[f1+f2]∆[f1+f2]((2pi)−3/2∗(q, λ)√2|~q|∫dx eiq·xδδh(x)ei∫[f1+f2]h|h→0),(3.32)with(f1|2)µν(w, u1|2) ≡12∑qe−iq·(w−u1|2)∫ ∞0ds[τ1|2µν eisp1|2·q + τ ′1|2µν e−isp′1|2·q].(3.33)Note that this result can also be written as(f1|2)µν(w, u1|2) ≡12∫ ∞0ds[τ1|2µν δ(w − u1|2 − sp1|2) + τ ′1|2µν δ(w − u1|2 + sp′1|2)],(3.34)443.2. Two-Body Scatteringrevealing that these functions are just the classical gravitational sources ofparticles traveling at constant velocity.3.2.3 Multiple Graviton EmissionThe generalization to n gravitons is the same as before, meaning∂An∂κ2=(2pi)−614√ω1ω′1ω2ω′2∫du1∫du2 ∆αβσρ(u1 − u2)τ(1)αβ τ(2)σρκ2× eiu1·(p′1−p1)eiu2·(p′2−p2)e i2∫[f1+f2]∆[f1+f2]×n∏m((2pi)−3/2∗(qm, λm)√2| ~qm|∫dxm eiqm·xm δδh(xm)ei∫[f1+f2]h|h→0),(3.35)and as before, the graviton part simplifies ton∏m 2∑j=1√8piG(2pi)3/2√2| ~qm|[µν∗m (p′j)µ(p′j)νp′j · q − iδ− µν∗m (pj)µ(pj)νpj · qm + iδ] ≡ n∏mκΩm,(3.36)but the factorization of the soft graviton dependence of A is not yet estab-lished - only for ∂A/∂κ2 so far. In order to argue that the κΩm factor at thelevel of the amplitude, we will specialize to the case of small virtual gravitonmomenta as well.3.2.4 Virtual Graviton ExchangeInside of exp(i2∫[f1 + f2] ∆ [f1 + f2])are terms like (i, j ∈ {1, 2}) pi∆pi,pi∆p′i, pi∆pj , etc. The pi∆pi terms again just lead to renormalization of asingle particle in the forward limit. The pi∆p′i terms contribute to renormal-izing the vertex, which was seen in the bremsstrahlung calculation, equation(3.26), but here we choose to drop them. The remaining cross terms will453.2. Two-Body Scatteringgenerate the ladder type graphs that are known to give the eikonal amplitudewhen summed [44]. Thus, we approximateei2∫[f1+f2]∆[f1+f2] → ei∫f1∆f2 . (3.37)If the momenta of the virtual radiation are small, we have that pi ≈ p′i, andwe also replace(f1|2)µν(w, u1|2)→12∑qe−iq·(w−u1|2)∫ ∞−∞ds τ1|2µν e−isp1|2·q=12∫ ∞−∞ds τ1|2µν δ(w − u1|2 − sp1|2),(3.38)so thatei∫f1∆f2 ≈ eiτ1αβτ2σρ∫∞−∞ ds1∫∞−∞ ds2 ∆αβσρ(u1−u2−s1p1+s2p2). (3.39)To evaluate the exponent, use the integral representation of ∆, repeatedhere for convenience,∆αβσρ(x) = limM→0∑keik·xk2 −M2Pαβσρ, Pαβσρ ≡ 12[ηασηβρ+ηαρηβσ−ηαβησρ],(3.40)where we have introduced a fictional graviton mass M in order to keep trackof divergences later. The limit will remain implicitly. We work in the center-of-mass frame, where p1 = (E, 0, 0, p), and p2 = (E, 0, 0,−p). The propertime integrals giveei∫f1∆f2 ≈ exp iκ2γ(s)16Ep(2)∑keik⊥·u⊥k2⊥ −M2 , (3.41)where γ(s) ≡ 2(p1 · p2)2 −m4 = 12(s − 2m2)2 −m4, to be consistent withthe notation in [44], and s is the usual Mandelstam variable, s ≡ (p1 + p2)2.For some four-vector v, we have defined v⊥ ≡ (v1, v2) ≡ (vx, vy), and u ≡u1−u2. We see then, that taking the eikonal limit has effectively reduced the463.2. Two-Body Scatteringproblem to one with a two-dimensional dependence (i.e., dependence on u⊥,rather than all of u). The two-dimensional integral can also be performed,givingiκ2γ(s)16Ep(2)∑keik⊥·u⊥k2⊥ −M2=iκ2γ(s)32piEpK0(Mu⊥) ≈ − iκ2γ(s)32piEpln(Mu⊥). (3.42)Then in the limit of small virtual momenta, inserting the same integralrepresentation of the other graviton propagator in (3.35) - the one not inthe exponent - the amplitude becomes 5∂An∂κ2= (2pi)−2δ(p1 + p2 − p′1 − p′2)[n∏mκΩm]14√ω1ω′1ω2ω′2×∫du⊥ eiu⊥·Q(8Ep∂∂κ2)exp iκ2γ(s)16Ep(2)∑keik⊥·u⊥k2⊥ −M2 (3.43)We will drop the δ-function henceforth. Q ≡ (p′1 − p1)⊥ parameterizes thetwo-dimensional momentum transfer.3.2.5 Soft FactorizationDisplaying the factorization property is now reduced to solving a differentialequation. Defining a viaAn ≡ (2pi)−2[n∏mΩm]8Ep4√ω1ω′1ω2ω′2∫du⊥ eiu⊥·Q an(u⊥, κ), (3.44)5Really, the δ-function showing up here should be δ(p1 + p2 − p′1 − p′2 −∑m qm), butto leading order in 1/q, this reduces to what appears in eq (3.43).473.2. Two-Body Scatteringa obeys∂an(u⊥, κ)∂κ2= κn∂∂κ2exp iκ2γ(s)16Ep(2)∑keik⊥·u⊥k2⊥ −M2 ≡ κn ∂∂κ2exp(iκ2f).(3.45)This equation has the solution [4]an(u⊥, κ) = κnΓ[n2 + 1,−iκ2f](−iκ2f)n/2 + const., (3.46)where Γ[a, x] is the incomplete Gamma function. However, recalling (3.42)we can use the fact that the graviton is massless, and that limM→0 f =+∞. We must also enforce the boundary conditions, which state thatlimκ→0An = 0, and that as n → 0, we recover the amplitude with nosoft radiation. Then to leading order, [3]an(u⊥, κ) = κn(eiκ2f − 1)+O(1f), (3.47)An = (2pi)−2[n∏mκΩm]8Ep4√ω1ω′1ω2ω′2∫du⊥ eiu⊥·Q[eiχ(s,u⊥) − 1],(3.48)in which,χ ≡ κ2f ≡ −κ2γ(s)32piEpln(Mu⊥) = −Gγ(s)Epln(Mu⊥), (3.49)in agreement with [44]. Finally, we have shown that the radiative two-body eikonal amplitude factorizes precisely as predicted by the soft theo-rems:An = Aeik.0 ×n∏m 2∑j=1√8piG(2pi)3/2√2| ~qm|[µν∗m (p′j)µ(p′j)νp′j · q − iδ− µν∗m (pj)µ(pj)νpj · qm + iδ](3.50)483.2. Two-Body ScatteringThe soft theorems are universal, in that they do not discriminate based onthe spin of the matter or the particular hard process. We have claimedhere that they are also encoded in the functional formalism in a naturalway. The key result of this section is the derivation of the soft factors fortwo concrete processes, gravitational bremsstrahlung and two-body eikonalscattering, with the latter being a new result of this formalism. The resultsare not surprising, however a few questions still remain. The cancellation ofinfrared divergences is straightforward in the bremsstrahlung case, followingclosely the pattern of [29], but the same cancellation has not yet been explic-itly demonstrated in the two-body scattering case. Additionally, we havethroughout this thesis simply stated things at the lowest contributing orderwhen expanding in small momenta. A great deal though has been said aboutsubleading soft effects [59–61] and they have been associated with new sym-metries and memory effects in both QED [62] and gravity [63, 64]. It seemsthat now that the functional formalism has been shown to neatly reproduceknown results at leading order, it would be a mistake not to attempt to takeit to subleading order as well. The scalar theory also has an extra source ofsubdominant contributions, namely the higher order terms in the expansion(2.20).49Chapter 4ConclusionThe aim of this work has been to demonstrate the utility of the functionalformalism in understanding the infrared limit of quantum gravity in a uni-fied language. To that end, we show here that it neatly reproduces manyimportant known results from the literature. This should be seen as a firststep in using this formalism to fully understand all universal properties ofsoft gravity, due to the ease of applying this formalism to many aspects ofthe IR theory.In this thesis we have introduced the major problems in IR gravitationalphysics, and solved them using functional techniques. Chapter 1 coveredthe example of quantum electrodynamics, and discussed how to constructa simple theory of low energy quantized Einstein GR minimally coupled toscalar matter. The theory was then investigated using functional methodsand nonperturbative results were obtained in the main body of the work.In Chapter 2, we applied the powerful Fradkin technique to the linearizedtheory, yielding a useful representation for the generating functional. Wethen demonstrated the ease at which correlation functions could be recov-ered from this expression. As an example, the infrared effective form ofthe propagator was derived and its form discussed. Chapter 3 contained arecipe for computing S-matrix elements with the help of the eikonal form ofthe generating functional and the two-point function derived from it. Wethen finally demonstrated that the soft graviton theorem is encoded in thisformalism by deriving in parallel the eikonal form of the two-body scatteringamplitude and the soft graviton factorization property.We have already begun to speculate on potential applications of this work50Chapter 4. Conclusionbeyond the scope of this thesis. Future investigations using this frameworkmay include applications to decoherence due to quantum gravity [58], anattempt at understanding “dressed state” type approaches [51–57], and therelation of the functional formalism to the “infrared triangle” and the blackhole information problem as presented in e.g. [46–50]. Additionally, somemore technical aspects of this formalism play a central role, but are some-what opaque and deserve to be better understood. 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Strominger, “Evidence for a new soft graviton the-orem,” arXiv:1404.4091 (2014).[64] S. Pasterski, A. Strominger, and A. Zhiboedov, “New gravitationalmemories,” J. High Energ. Phys. 2016: 53 (2016).[65] D.J. Gross, R. Jackiw, “Low-energy theorem for graviton scattering,”Phys. Rev. 166 1287 (1968).[66] R. Jackiw and L.D. Solov’ev, “Low-energy-theorem approach to single-particle singularities in the presence of massless bosons,” Phys. Rev.173 1485 (1968).[67] J. Schwinger, “Sources and electrodynamics,” Phys. Rev. 158 1391(1967).57Appendix ASelected FunctionalIdentitiesHere we state without proof some necessary functional identities.The action of an operator linear in functional derivatives on any functionalF of some function j can be shown to bee∫f δδjF [j] = F [j + f ], (A.1)simply shifting the functional dependence of F .We also need to know the action of a quadratic shift on certain types offunctionals. The quadratic shift operator is of the form, e.g.,e− i2∫δδjA δδj (A.2)and we have the special casese− i2∫δδjA δδj ei∫fj = ei2∫fAfei∫jf , (A.3)when acting on a linear functional, ande− i2∫δδjA δδj ei2jDj = exp{i2∫jD(1−AD)−1j − 12Tr ln(1−AD)}, (A.4)acting on a Gaussian functional.Another identity that is extremely useful for evaluating S-matrix elements58Appendix A. Selected Functional Identitiesin this formalism ise−i ∫ f δδj ei2∫j∆j = ei2∫j∆je−i2∫δδh∆ δδh e∫fh, (A.5)where h ≡ ∫ ∆j.59Appendix BEvaluation of PhaseIntegralsThe exponent in eq. (2.45) has a phase∑qf(q)f(−q)ταβ(k, k)∆αβµν(q2)τµν(k, k) = ταβPαβµντµν∑qf(q)f(−q)q2≡ ταβPαβµντµνI,(B.1)where Pαβµν ≡ 12 [ηαµηβν + ηανηβµ − ηαβηµν ]. Now using the expression forthe form factor in (2.28), we can approach I in a simple way. First, write Iusing the unintegrated expressions for the form factors:I =−∫ s0ds′∫ s0ds′′∑q1q2e−i(s′−s′′)(2k·q)≡−∫ s0ds′∫ s0ds′′∑q1q2e−is−(2k·q)(B.2)This expression is ∼ ∫∞0 QdQ and thus still has a divergence in the UV.However, we can move this divergence out of the momentum integrals andinto the proper time ones by introducing yet another proper time (and goingto Euclidean momentum space)I = −∫ s0ds′∫ s0ds′′∫ ∞0du∑qeiuq2−is−(2k·q). (B.3)60Appendix B. Evaluation of Phase IntegralsNow the momentum integral is Gaussian and can be done, givingI = − i4pi2∫ s0ds′∫ s′0ds′′∫ ∞0duu2exp(−ik2s2−u), (B.4)The integral over u is elementary when s2− is given a small imaginary part:s2− → s2−− i. Actually it is more convenient to take s− → s−− i, which isequivalent in the limit of small since s′′ < s′. The result isI = − i4pi2k2∫ s0ds′∫ s′0ds′′(s− − i)−2, (B.5)and with a little bit of algebra this can be shown, to leading order, to behaveasI = i4pi2k2[isΛ− ln(sΛ) +O(Λ0)] , (B.6)where Λ ≡ −1 may be identified as the UV cutoff avoided in the momentumintegrals earlier.61Appendix CThe Generating FunctionalThis appendix and the next deal with deriving eq. (3.5). We closely followthe arguments given in chapters 3 and 4 of [1].In classical probability theory, the expectation value of some observable Othat depends on a random variable x is given by〈O〉 =∫dxP (x)O(x), (C.1)where P (x) is the probability of some value of x, with∫dxP (x) = 1. Quan-tum field theory is similar, in that we typically consider expectation valuesof, e.g., field operators at different spacetime points,〈φ(x1) · · ·φ(xn)〉 =∫Dφ(x) P[φ]φ(x1) · · ·φ(xn). (C.2)Here, P[φ] = eiS[φ] is the probability amplitude for the field configurationφ(x). We can consider the following expectation value:Z[j] ≡ 〈z[j]〉 ≡ 〈T ei∫φj〉 (C.3)with T the time ordering operator. Writing out the functional average,Z[j] = N∫Dφ eiS[φ]+i∫φj , (C.4)we see that this is just the usual generating functional, which generatescorrelation functions in a particular theory (this formalism extends triviallyto multiple interacting fields). N−1 = ∫ Dφ eiS[φ], to ensure that Z[j = 0] =62Appendix C. The Generating Functional〈0|0〉 = 1 (using the IN state basis). However, we can learn a bit more byinvestigating z instead of just Z. We generalize z by definingzba[j] ≡ T ei∫ ba φj , (C.5)where now∫ ba φj ≡∫d3x∫ tbtadt φ(x)j(x). Then,δzbaδj(x)= i zbx φ(x) zxa, (C.6)by time ordering, or for z ≡ z ∞−∞,δzδj(x)= i z∞x φ(x) zx−∞. (C.7)This property of z will allow a connection with the S-matrix.63Appendix DThe Generating FunctionalGenerates the S-MatrixFirst define the usual asymptotic IN/OUT states,|a〉OUT = limt→∞ |a, t〉 , |a〉IN = limt→−∞ |a, t〉 (D.1)and the S-matrix via|a〉OUT = S† |a〉IN , φOUT(x) = S†φIN(x)S. (D.2)Generally, the stationary action principle enforces an on-shell relation likeDˆxφ(x) = j(x), (D.3)where Dˆ is some differential operator acting w.r.t. the coordinate x, and j(x)is a spacetime-dependent source. The general solution to this equation canbe expressed in terms of the Green function G(x, x′) satisfying DˆxG(x, x′) =δ(x− x′) asφ(x) = φ0(x) +∫dy G(x, y)j(y) = φ0(x) +∫dy G(x, y)Dˆyφ(y), (D.4)or in terms of the IN/OUT fields,φ(x) = φ INOUT(x) +∫dy GRA(x, y)Dˆyφ(y), (D.5)64Appendix D. The Generating Functional Generates the S-MatrixWhere R (A) denotes the retarded (advanced) propagator. By putting (D.5)into (C.7), we get thatδzδj(x)= i z φIN(x) +∫d4y GR(x, y)Dˆy δzδj(y)= i z φOUT(x) +∫d4y GA(x, y)Dˆy δzδj(y).(D.6)It does not matter whether we choose to express this using the IN or OUTfields. Take the difference of the two equivalent expressions to see that0 = i (φOUT(x)z − zφIN(x)) +∫d4y G(x, y)Dˆy δzδj(y), (D.7)G now meaning the causal propagator. Multiply this expression by S anduse (D.2) to show that[φIN(x),Sz] = iS∫d4y G(x, y)Dˆy δzδj(y), (D.8)which can be rewritten as[φIN(x),Sz] = i∫d4y G(x, y)Dˆy δδj(y)(Sz), (D.9)because S will not depend explicitly on j(x). This relation is not immedi-ately useful, but we will see that is enough to determine the form of Sz. Todo this, consider another operator similar to z, but with normal orderingrather than time ordering. The normal ordering operation is denoted by ::,and demands that, for everything between the colons, creation operators areplaced to the left of annihilation operators. E.g.,: aˆ†aˆ : = aˆ†aˆ, : aˆaˆ† : = aˆ†aˆ, (D.10)etc. We consider the operator: e∫φINf :, (D.11)65Appendix D. The Generating Functional Generates the S-Matrixand make use of the result[φIN(x), : e∫φINf :]= : e∫φINf : i∫d4y G(x, y)f(y). (D.12)Looking at this as well as (D.9), it appears thatSz = : e∫φINDˆ δδj : g[j], (D.13)for some g[j]. In fact, this form is fixed as well from the fact that 〈: e∫φINf :〉 =1, and we getSz[j] = : e∫φINDˆ δδj : 〈Sz[j]〉 . (D.14)Finally, if the source j(x) is set to zero after the functional derivatives aretaken, we isolate the form of S up to normalization:S = : e∫φINDˆ δδj : Z[j]|j=0, (D.15)where again, Z ≡ 〈z〉. The argument proceeds in the same way for theorieswith multiple fields.66