UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Manifestly gauge invariant transition amplitudes and thermal influence functionals in QED and linearized… Wilson-Gerow, Jordan 2017

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2017_september_wilson-gerow_jordan.pdf [ 608.09kB ]
Metadata
JSON: 24-1.0354259.json
JSON-LD: 24-1.0354259-ld.json
RDF/XML (Pretty): 24-1.0354259-rdf.xml
RDF/JSON: 24-1.0354259-rdf.json
Turtle: 24-1.0354259-turtle.txt
N-Triples: 24-1.0354259-rdf-ntriples.txt
Original Record: 24-1.0354259-source.json
Full Text
24-1.0354259-fulltext.txt
Citation
24-1.0354259.ris

Full Text

Manifestly Gauge Invariant TransitionAmplitudes and Thermal InfluenceFunctionals in QED and LinearizedGravitybyJordan Wilson-GerowB.Sc., The University of Victoria, 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© Jordan Wilson-Gerow 2017AbstractEinstein’s theory of General Relativity tells us that gravity is not a forcebut rather it is the curvature of spacetime itself. Spacetime is a dynamicalobject evolving and interacting similar to any other system in nature. Theequivalence principle requires everything to couple to gravity in the sameway. Consequently, as a matter of principle it is impossible to truly isolate asystem—it will always be interacting with the dynamical spacetime in whichit resides. This may be detrimental for large mass quantum systems sinceinteraction with an environment can decohere a quantum system, renderingit effectively classical. To understand the effect of a ‘spacetime environ-ment’, we compute the Feynman-Vernon influence functional (IF), a usefultool for studying decoherence. We compute the IF for both the electromag-netic and linearized gravitational fields at finite temperature in a manifestlygauge invariant way. Gauge invariance is maintained by using a modifica-tion of the Faddeev-Popov technique which results in the integration overall gauge equivalent configurations of the system. As an intermediate stepwe evaluate the gauge invariant transition amplitude for the gauge fieldsin the presence of sources. When used as an evolution kernel the transitionamplitude projects initial data onto a physical (gauge-invariant) subspace ofthe Hilbert space and time-evolves the states within that physical subspace.The states in this physical subspace satisfy precisely the same constraintequations which one implements in the constrained quantization method ofDirac. Thus we find that our approach is the path-integral equivalent ofDirac’s. In the gauge invariant computation it is clear that for gauge the-ories the appropriate separation between system and environment is not a)matter and gauge field, but rather b) matter (dressed with a coherent field)and radiation field. This implies that only the state of the radiation fieldcan be traced out to obtain a reduced description of the matter. We stressthe importance of gauge invariance and the implementation of constraintsbecause it resolves the disagreement between results in reported recent lit-erature in which influence functionals were computed in different gaugeswithout consideration of constraints.iiLay SummaryThe macroscopic world is described well by classical physics. At a momentof time you say i) where things are, ii) how fast they are moving, andiii) how hard are they being pushed or pulled. This is all you need tosay where everything will be at a later time and how fast it will all bemoving. This description works well until you look at very small things likeelectrons. We’ve seen experimentally that small things are better describedby quantum mechanics. In quantum mechanics things are more strange.For example, an object can be in more than one place at a time. Since wedon’t see macroscopic objects behaving this way, we expect there to be somesort of cross-over. In this thesis we build some mathematical tools whichwill help to understand how fluctuations in the gravitational field may beresponsible for this cross-over.iiiPrefaceThe idea to compute a gravitational influence functional was provided bythe author’s supervisor Dr. Philip Stamp. Aside from this initial idea, thework done in this thesis was performed independently by the author.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Previous Approaches to Gravitational Decoherence . . . . . 41.1.1 Decoherence in Modified Quantum Theories . . . . . 51.1.2 Decoherence in Conventional Quantum Theory . . . . 61.2 Quantum Gravity as a Low Energy Effective Field Theory . 101.3 Feynman-Vernon Influence Functional . . . . . . . . . . . . . 121.4 Difficulty Defining an Influence Functional in a Gauge Theory 161.5 Quantum Gravity Path Integral - Boundary Terms . . . . . . 181.5.1 Gibbons-Hawking-York Boundary Term . . . . . . . . 181.5.2 Gauge Transformation Boundary Term . . . . . . . . 202 Transition Amplitudes in Gauge Theories . . . . . . . . . . 232.1 Quantum Electrodynamics (QED) . . . . . . . . . . . . . . . 242.1.1 Free U(1) Theory . . . . . . . . . . . . . . . . . . . . 242.1.2 Interacting QED . . . . . . . . . . . . . . . . . . . . . 292.2 Linearized Quantum Gravity . . . . . . . . . . . . . . . . . . 343 Influence Functional in QED and linearized Gravity . . . . 433.1 Thermal Photon Bath Influence Functional . . . . . . . . . . 433.2 Thermal Graviton Bath Influence Functional . . . . . . . . . 453.3 Importance of Gauge Invariance . . . . . . . . . . . . . . . . 48vTable of Contents4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55AppendicesA The Classical Action in QED and Linear Gravity . . . . . 63B Instantaneous Interaction Terms . . . . . . . . . . . . . . . . 66C Derivation of the Master Equation from the Path Integral 69viAcknowledgementsThe author would like to thank Daniel Carney, Laurent Chaurette, YanbeiChen, Colby DeLisle, Michael Desrochers, Dominik Neuenfeld, and DavidStephen for helpful discussions. Special thanks is extended to Philip Stampfor both suggesting a project on gravitational decoherence and sharing awealth of knowledge on the topic.viiDedicationTo my mother,For the unconditional support she has provided which has allowed meto pursue my academic dreams, and for teaching me that excellence onlycomes through unfailing dedication and hard work.viiiChapter 1IntroductionQuantum systems are fragile and easily disrupted by interactions with en-vironmental disturbances. Decoherence can occur when an environmentdisturbs a quantum system even if energy is not exchanged [1]. It is thesame thing that makes quantum systems so rich that makes them so frag-ile, they can build extraordinarily complex correlations. Counterintuitiveas it may be, in general an environment decoheres a quantum system notby physically destroying it but by developing intricate correlations with thestate of the system. For a quantum system strongly correlated with its en-vironment, a demonstration of its quantum behaviour requires the carefulcontrol of the state of the environment in addition to the system. Thus ifthe state of the environment is not carefully tracked then the system alonemay not exhibit quantum behaviour (interference for example). Quantumcorrelation, referred to as quantum entanglement, was initially thought ofas a demonstration of the incompleteness of quantum theory [2] but is nowunderstood as such a fundamental feature of nature that there is an entirefield based around the idea that it might be the glue which holds spacetimetogether [3].To study quantum systems experimentally we try to isolate quantumsystems to avoid environmental interactions, but as a matter of principleno system can be entirely isolated. Einstein’s theory of General Relativity(GR) has taught us that gravity should not be thought of as a force betweenobjects but rather as a manifestation of the curvature of spacetime itself.Spacetime is no longer a fixed background on which objects interact, it toois a dynamical object. Since it provides the casual structure for all otherobjects in nature, spacetime is necessarily interacting with everything;nothing can be shielded from gravitational interaction. Spacetime itself isan ever present environment which may cause the decoherence of quantumsystems.Gravitational decoherence is a hard problem to properly define let alonestudy and solve. In the typical discussion of decoherence one has two quan-tum systems, commonly referred to as the central system and the envi-ronment. A measurement/experiment involving only the central system is1Chapter 1. Introductioninsensitive to not only the state of the environment but also the correlationsthat exist between the environment and central system. Operationally thisignorance is claimed by summing (or tracing) over all possible states of theenvironment. Any information stored as correlations between the centralsystem and environment is lost, i.e. the entropy of the central system is in-creased. It is then said that the central system has undergone decoherence.In quantum mechanics (QM) alone this leads to a number of unresolvedquestions related to the interpretation of the theory. For example, whatdefines a measurement? Does the wavefunction/density matrix simply rep-resent the information we have, or is it something physical? Typically suchquestions are set aside because we can still use QM to predict probabili-ties of and correlations between measurement outcomes without thinkingabout the ontology of measurement operators and state vectors. This is theFAPP (‘for all practical purposes’) approach, which has remained popularbecause of how accurate quantum mechanical predictions can be regardlessof the chosen interpretation. Note, for example, the agreement betweentheory and experiement of the anomalous magnetic moment of the elec-tron to better than 1 part in 1011 [4]. However once gravitation is includedthe foundational questions which could once be ignored FAPP now becomeessential problems which make it unclear what/how computations can bedone in quantum gravity. For example, if spacetime is treated as a quantummechanical object then what constitutes an observer/measuring apparatus?Even if spacetime is treated classically we can ask, if a central system in-teracts with an environment that later gets thrown into a black hole, arethe correlations permanently lost? The latter question seeds the well knowblack hole information paradox [5]. Questions of this type make it difficultto formulate meaningful questions involving QM and gravity. The problemis that we simply do not yet have a complete accepted theory incorporatingboth quantum mechanics and general relativity.There is not yet a unification of GR and QM, however there are a numberof different approaches to this unification. In the most popular approaches,string theory and loop quantum gravity, QM holds at all scales howeverat short distances the classical notion of spacetime is drastically modified,i.e. general relativity breaks down. There are a number of alternativeapproaches in which QM suffers a breakdown at macroscopic scales as aresult of gravitation. These theories predict new mechanisms which canlook like intrinsic decoherence of quantum states (as opposed to “environ-mental decoherence” caused by interaction with environmental degrees offreedom) [6–14]. For compatibility with previous experimental observationsthe predicted decoherence rates must be negligible at microscopic scales,2Chapter 1. Introductionbut they are expected to become appreciable around mesoscopic scales andto completely suppress quantum effects on macroscopic scales. Recentlythere has been much effort towards the development of “table-top” quan-tum gravity experiments which aim to study quantum theory at mesoscopicscales right where intrinsic decoherence mechanisms are expected to be seen.To understand the results of such experiments it is essential to understandhow environmental decoherence may occur in conventional quantum gravity(theories without modifications to QM). We can see the extreme sensitivityto decoherence as providing an useful probe of quantum gravity. In this the-sis, we develop tools which will be useful for the study of decoherence in theeffective field theory description of conventional quantum gravity. Specif-ically we compute the Feynman-Vernon influence functional for a bath ofgravitons at finite temperature. As a warm-up for quantum gravity we firststudy Quantum Electrodynamics (QED)1. As we will show, in the low en-ergy limit which we are interested in the treatments of the two theories arequite similar.This thesis is organized as follows. This chapter provides an introductionto the study of gravitational decoherence. In Section 1.1 we provide a briefdiscussion of some previous work on gravitational decoherence in: classicalgravity, gravitationally induced intrinsic decoherence theories, and conven-tional quantum gravity theories. In Section 1.2 we discuss how althoughwe lack a full theory of quantum gravity we can still make quantitativepredictions in a low energy limit. In Section 1.3 we provide a general intro-duction to the Feynman-Vernon influence functional, the primary tool forunderstanding decoherence in the path-integral formalism. In Section 1.4 wediscuss the additional subtleties in formulating an influence functional forgauge theories. In Section 1.5 we discuss some subtleties regarding boundaryterms in quantum gravitational path integrals. In Chapter 2 we introducethe gauge theories which we will be studying. En route to computing theinfluence functional, we compute the gauge invariant propagator for thesegauge theories using a modification of the FP trick applied to transitionkernels. It must be understood that we are using the word propagator tomean the amplitude for the system to evolve from one configuration to an-other configuration in a given time, i.e. the Feynman transition kernel. Theword propagator in field theory has become synonomous with correlator,but it should be clear that we are not discussing correlation functions. For1By QED we are not specifically referring to the theory involving Dirac fermions cou-pled to a U(1) gauge field but rather a quantum theory of a generic not-yet-specifiedmatter which is minimally coupled to a U(1) gauge field. Of course this includes Diracfermions but also e.g charged scalar fields and point particles.31.1. Previous Approaches to Gravitational Decoherenceinstructive purposes we’ll first show how such a computation is done forfree U(1) theory, then for QED, and ultimately for linearized gravity withmatter. In Chapter 3 we compute the influence functional for both QEDand linearized gravity at finite temperature. We also discuss the result onewould obtain if the gauge invariance of these theories is not properly treatedand show that this matches a result reported in recent literature [15]. All ofthese discussions and results are summarized in the conclusion, Chapter 4.Supplementary materials are presented in the appendices.Throughout we will use units in which (h¯ = c = kB = 1), and a mostlypositive metric convention (− + ++). We will use the shorthand notation∫ tftid4x =∫ tftidt∫d3x. The letters i, f will be reserved to label initial andfinal quantities. They will never be used as indices. We will use Greekletters for spacetime indices and Latin letters in the middle of the alphabet(except i, f) for spatial indices e.g. (j, k, l). Latin letters at the start of thealphabet e.g. (a, b, c) will be used to label miscellaneous discrete quantities.This being said, we will often suppress spacetime indices as well as thearguments of functions to avoid cluttered expressions. It should be clearfrom the context which objects are vectors and tensors.1.1 Previous Approaches to GravitationalDecoherenceGravitational decoherence is a massive topic which receives both theoreticaland experimental interest from communities ranging from quantum infor-mation to opto-mechanics to quantum cosmology. The idea that gravitationmay be responsible for the quantum to classical transition was present anddiscussed in the early Sixties. Comments can be found in the Feynmanlectures on gravitation [16] as well as in the work of Rosen [17]. The firstactual model was soon after proposed by Karolyhazy [6]. Of course, sincewe lack a full theory of quantum gravity, our understanding of gravitationaldecoherence is still incomplete and it remains an actively studied topic.In this section we will attempt to provide a bit of an overview of thedifferent ways in which gravitational decoherence has been studied. Ouraim is not to provide a comprehensive review but rather to help situateour work in the appropriate historical context. More detailed summaries ofprevious works are provided in [14, 18–20].41.1. Previous Approaches to Gravitational Decoherence1.1.1 Decoherence in Modified Quantum TheoriesAs previously mentioned, theories incorporating both quantum mechanicsand gravity can separated into two groups: those which hold QM sacred,and those willing to violate QM at macroscopic scales. Of the theories whichviolate QM, there are different approaches distinguished by how gravitationenters the theory.The earliest ideas of this kind, as well as those popularized by Penrose in-volve taking the Heisenberg uncertainty relations for quatum systems livingon a spacetime manifold and understanding this as leading to an inherentfuzziness to spacetime. This fuzziness leads to the loss of phase relationsin the different branches of the matter wavefunction and thus a pure stateevolves into a mixed state [6, 7]. In these theories decoherence is an intrinsicprocess in nature.Another class of theories assume that gravity is inherently classical butQM is necessarily modified. In particular Mielnik [21] emphasized that,‘either the gravitation is not classical or quantum mechanics is not orthodox’.These theories view QM as a linear approximation to a more complicatednon-linear theory where the non-linearities arise from gravitation [8–10].This leads to a modification to the Einstein field equation where the RHSof the equation is replaced by the expectation value of the stress-energytensor operator. The non-linear nature of these theories leads to a dynamicalcollapse of the wavefunction which proceeds most rapidly for macroscopicobjects. While this is not necessarily decoherence, the inability to sustainmacroscopic superpositions is a feature common to both decoherence modelsand collapse models. Without proper attention to detail an observation ofdynamical collapse could be misinterpreted as environmental decoherence.In another approach the graviational field is treated as a classical fluc-tuating stochastic variable. This is an example of a stochastic collapse the-ory. Stochastic collapse theories propose that some new universal mecha-nism is responsible for suppressing macroscropic quantum fluctuations [22].They have been well studied outside of a gravitational context as a potentialmeans for understanding macrorealism and the collapse of the wavefunction(see [19] and references therein). The idea that the underlying mechanismmay be related to gravitation was popularized by Dio´si [11, 12]. In these the-ories the corresponding evolution equation for the matter density matrix isno longer the unitary von Neumann equation, but instead it is a Markovianmaster equation describing the decoherence of pure states. Dio´si’s approachis limited for a number of reasons, the most obvious of which being that itis not relativistic. In addition the theory has been criticized by Ghirardi et51.1. Previous Approaches to Gravitational Decoherenceal who claim it is not internally consistent [13].The final intrinsic decoherence theory we’ll describe is that of correlatedworldlines (CWL theory) [14]. In CWL theory the linearity of QM is brokenin the path integral. In a conventional path integral computation one sumsup all paths (independent of one another) which a system can take to evolvefrom one configuration to a later configuration. In CWL theory it is assumedthat the conventional path integral is the lowest order term in an infiniteseries of multiple path integrals. The second order term includes a sum overall pairs of paths, the third order goes over all triplets, and the n-th orderover all n-tuplets of paths. What distinguishes CWL theory from the con-ventional path integral is that the paths are no longer independent from oneanother, they are correlated. Arguments based on the equivalence principleand indistinguishability in QM suggest that the correlations are generatedby gravitation. This leads to “path-bunching” where paths “gravitate” to-wards each other. This suppresses quantum fluctuations for macroscopicobjects and thus predicts a quantum-to-classical crossover. This too is atheory without decoherence, but again the experimental signature of thiscrossover may be misattributed to environmental decoherence if we do nottake care to understand both effects well.It is clear then that we can no longer naively attribute decoherence inexperiments to dirt/noise. Experimental signatures of the above theoriesmay already be lying in our data, but we don’t yet understand conventionalgravitational decoherence well enough to interpret the data and place boundson these theories. Since the Planck energy scale is far beyond present theday energy frontier we cannot afford to dismiss potential experimental datawhich probes theories of quantum gravity. In the next section we will reviewsome efforts to understand and model decoherence in conventional quantumgravity.1.1.2 Decoherence in Conventional Quantum TheoryWithout a full theory of quantum gravity, the various approaches to deco-herence in conventional quantum gravity have been rather ad hoc. Assump-tions must be made about how to correctly describe spacetime, and manydifferent choices have been made. In most of the approaches we will discussthe environment consists of small fluctuations about a classical backgroundspacetime metric (typically Minkowski, gµν = ηµν + hµν). Without fixing aclassical background it is unclear how to even describe the central system—how can we possible describe a particle in a superposition of two locationsin a coordinate-independent way [7]? It has been argued that diffeomor-61.1. Previous Approaches to Gravitational Decoherencephism invariance is so strict that there are no local observables in quantumgravity; the only observables are defined on the boundary of the spacetimemanifold [23, 24]. This is exemplifed by the ADM mass being a boundaryintegral [25]. Indeed this is the spirit of the holographic principle [26, 27],and is realized by the AdS/CFT correspondence [28–30]. Until we furtherdevelop our understanding of quantum gravity we must assume a classicalbackground on which the central system and gravitational fluctuations canlive. This being said, since the gravitational fields created in any man-madeexperiment will certainly be in the weak-field regime we can hypothesizethat this assumption is reasonable for all practical purposes.Before we discuss the many approaches which are similar to ours (con-sidering only small spacetime fluctuations) its worth mentioning some morespeculative ideas which do not involve a static classical background metric.It was hypothesized that even conventional quantum gravity is inherentlynon-unitary. This was an idea which recieved a lot of attention from Hawk-ing as a potential resolution to the black hole information paradox. Some ofhis ideas included: thermalization due to the formation and evaporation ofblack holes [5], metric fluctuations destroying global hyperbolicity [31], andthe branching off of “baby universes” connected to ours by wormholes [32–34]. It is notable that the wormhole computations suggest a crossover massscale above which macroscopic objects would rapidly decohere, but this cal-culation was not performed within a controlled approximation and thus itslegitimacy is questionable.Recent attention has been given to the idea that even static classicalgravitational fields may cause quantum decoherence. This idea was pro-posed by [35] and has since been a hotly debated topic [36–40] (see also [20]and refs. therein). The basic setup for this idea is a single object consist-ing of many microscopic constituents (for example a large molecule). Thedegrees of freedom of the system decompose into center-of-mass motion andinternal excitations. The claim in [35] is that the center-of-mass motionbecomes coupled to the internal degrees of freedom due to the gravitationaltime dilation felt by the internal modes based on the trajectory of the center-of-mass. The internal modes act as “internal clocks” and it is argued that ifthese clocks register different proper time durations then they can no longerconstructively interfere. The gravitational field does not act as the environ-ment but rather the internal modes constitute the environment while thecenter-of-mass degree of freedom is the central system. This is an inter-esting idea, but since it is still hotly debated (and a different flavour thanthe gravitational decoherence we are interested in) we will not have more tocomment on it.71.1. Previous Approaches to Gravitational DecoherenceIn the more common approaches a classical metric is fixed (typicallyMinkowski) and the effect of linearized metric fluctuations on a central sys-tem are studied. Such fluctuations occur in classical gravity where onecan assume that astrophysical or cosmological processes produce a stochas-tic background of classical gravitational waves [41, 42]. More interestingto the quantum gravity community is the effect of quantum fluctuationsassociated with the zero-point motion of the metric field. The quantumfluctuations have been modelled in a number of different ways. Many havetreated the metric perturbation as a stochastic variable with a gaussianprobability distribution. This was done for a non-relativistic particle cou-pled to fluctuations of the conformal factor in [43], i.e. the scalar φ inthe metric expansion hµν = (2φ + φ2)ηµν . Such a metric ansatz does notsatisfy Einstein’s equations, but this was soon after remedied by includingshearing modes [44]. Isotropic pertubations were well studied, and it wasshown that a perturbation of the form hij = ξ(t)δij with mean 〈ξ(t)〉 = 0,〈ξ(t)ξ(t′)〉 = M−1P δ(t − t′) would lead to decoherence into the energy basiswith a rate Γ ∼ (∆E)2/MP [45, 46]. Here ∆E is the difference in energy ofthe two states under consideration in the off-diagonal element of the densitymatrix. In all of these models the correlation functions for the noise were notderived but simply assumed. Their models contained free parameters whichwere identified with the Planck constants of appropriate units (mass, time,etc.). In the papers of Ford [47, 48], care was taken to perform a properQFT computation of the transverse-traceless graviton thermal corrleationfunction for use in a stochastic model. The novelty in this model is that thequantum particle was not coupled directly to the gravitons but rather thewalls confining the particle were. The background fluctuations then led tofluctuations in the size of the container causing decoherence for the parti-cle inside. Ford predicts a decoherence rate arising from interaction with agraviton bath at temperature T , Γ ∼ T (∆E/MP ). This depends only lin-early on the energy difference and, in contrast to the previous computations,it vanishes at zero temperature.The above models attempted to describe the quantum fluctuations ofthe metric by treating it as a classical fluctuating variable. This can providesome intuition but to truly directly understand quantum gravitational effectswe need to treat the metric perturbation quantum mechanically. This wasdone in [15, 49–53]. In each of these references a decoherence rate for asample matter system was either computed or estimated. It is difficult todirectly compare the rates though because the different authors consideredquite different matter systems, e.g. massive scalar coherent states, pointparticles in one-dimension, a point particle in an interferometer, and photon81.1. Previous Approaches to Gravitational Decoherencewavepackets. We will later see that our computation is sufficiently generalthat all of these examples could in principle be studied.In each reference the computation was done differently and not all re-sults agree quantitatively or even qualitatively. We’ll have more to say inChapter 3 about specifically how the results disagree and why this arises.In all cases the Einstein-Hilbert action is linearized and the metric pertur-bation is quantized either using the path-integral or canonical quantization.The initial state of the metric pertubation and matter is assumed to beuncorrelated. In all but [49] (where the initial state is vacuum) the metricperturbation is assumed to be thermal. In references [15, 49, 50, 52] thepath-integral is used and the effect of the environment is captured in aninfluence functional. The central system in [49, 50] is restricted to be a col-lection of non-relativistic point particles. In [52] the generalization is madeto general matter coupling to the metric perturbation via the stress-energytensor Lint ∝ hµνTµν . In each of [49, 50, 52] transverse-traceless (TT) gaugeis assumed altough this is a gauge choice which cannot be consistently madein the presence of matter [54]. Blencowe [15] chose an initially thermal stateand used the harmonic gauge fixing term (a valid choice), however there wasa drastic overcounting of the gravitational degrees of freedom. Although theTT gauge choice cannot be consistently made in the presence of matter itstill remains true that the two TT polarizations are the only independentdegrees of freedom in the metric perturbation; the remaining componentsof hµν are constrained variables. We will have much more to say aboutthis in the upcoming chapters, but for now it suffices to say that there hasnot yet been a satisfactory quantum mechanical derivation of the influencefunctional describing a thermal bath of gravitons. Certainly at high tem-peratures this should be well approximated by the classical results, but aproper QM calculation would allow one to interpolate between quantum andclassical regimes.In the other two references [51, 53] the metric perturbation is quan-tized canonically. They both make use of the ADM (3 + 1) decompositionto describe the time evolution of a spatial 3-metric hij [25]. In the ADMformalism (even at the linear level) one must explicitly deal with the con-straints of general relativity. In [51] a gauge is fixed and the constraints areimposed on the field operators, whereas in [53] they use Dirac constrainedsystem formalism [55, 56] to implement the constraints on the states anddescribe the dynamics in a manifestly gauge invariant way. In both cases theconstraints are properly treated and as a result only the two independentgraviton degrees of freedom of the metric act as an environment. They bothassume an initially thermal state for the gravitons, and they both compute91.2. Quantum Gravity as a Low Energy Effective Field Theorya master equation describing the evolution of the reduced density matrix forthe matter. It was not clear whether the results of [51] were gauge invariant,but because the Dirac formalism is manifestly gauge invariant there is noconcern about the results derived in [53].From the proper treatment of the constraints of the theory one learnsthat it is not enough for a decoherence model to be relativistic. To quote [51],since the Hamiltonian and momentum constraints of GR generate gaugetransformations which correspond to temporal and spatial reparameteriza-tions, “Any postulate of dynamical or stochastic fluctuations that correspondto space and time reparameterizations conflicts with the fundamental symme-tries of GR”. This comment has direct bearing on all of the modified QMtheories, the convential theories with fluctuations modelled as stochasticvariables, as well as the fully quantum theories. In particular this statementindicates that the results of Blencowe [15] cannot be correct since their resultwas obtained by integrating over fluctuations of all components of the met-ric perturbation hµν . In Section 3.3 we explicity show how Blencowe’s resultis obtained if the constraints/gauge-invariance of theory are not properlyaccounted for.It then seems as if the problem is solved and all that needs to be doneis to analyze the master equation derived in [53]. In principle this may becorrect but their master equation cannot be solved in general and it is notimmediately obvious how to implement different approximations. This iswhy an influence functional is powerful—the path integral approach offersnew approximation schemes (semiclassical, eikonal, etc.). This is why we areinterested in computing the gravitational influence functional in a way whichsatisfies the constraints of the theory and is manifestly gauge invariant.1.2 Quantum Gravity as a Low Energy EffectiveField TheoryAs previously mentioned, both string and loop approaches predict modi-fications to GR at high energies. If we assume quantum theory holds atall scales then there is a simple argument to suggest a breakdown of GRat high energies. Einstein’s theory of general relativity is described by theEinstein-Hilbert action2SEH =M2P2∫d4x√|g|R, (1.1)2This excludes boundary terms. Such terms do not modify the current discussion butwe will later discuss how they are essential for a number of other reasons.101.2. Quantum Gravity as a Low Energy Effective Field Theorywhere MP = (8piG)−1/2 is the Planck mass, g is the determinant of the met-ric tensor, and R is the Ricci scalar. This action is extremized by a spacetimemetric satisfying Einstein’s equation. Classical general relativity has beenextraodinarily well tested (see [57–59] and references therein). Indeed the re-cent discovery of gravitational waves emitted from a black hole merger eventhas verified a long standing prediction of classical general relativity [60] andhas opened up a new paradigm in strong-field testing [61]. One can chooseto describe the metric in terms of a deviation from the Minkowski metricgµν = ηµν + hµν/MP , and expand the Einstein-Hilbert action as an infiniteseries in hµν/MP . The result is the gauge theory of a massless spin-2 parti-cle (graviton) with an infinite number of interaction terms [62]. Every termwith n factors of h will be multiplied by M2−nP . The lowest order terms inthe expansion n = 2 have no powers of the Planck mass and simply look likestandard kinetic terms for a relativistic field L0 ∼ ∂h∂h. In principle onecan take this theory and try to perform standard perturbative quantum fieldtheory (QFT) computations using Feynman diagrammatics [63–65]. This isstraightforward until loop diagrams are considered and one must contendwith the fact that the theory is non-renormalizable.As in perturbative treatments of other quantum field theories the vari-ous ultraviolet divergences that arise from loop diagrams are absorbed intorenormalized coupling constants and physical observables can be computedin terms of these renormalized couplings. Of course the value of the renor-malized couplings must still be measured experimentally. In a renormaliz-able theory all ultraviolet divergences can be absorbed into a finite numberof renormalized coupling constants, and only a finite number of experimentsneed to be performed before the theory can make unambiguous predictions.Perturbative quantum GR is a non-renormalizable theory though; to ac-count for all ultraviolet diverences an infinite number of counterterms mustbe added, and thus infinitely many experiments must be performed to de-termine all of the the renormalized couplings [66, 67]. These countertermsextend beyond the Einstein-Hilbert action, in fact they include higher powersof the Riemann tensor. Naively this spells disaster for the theory, suggestingit has no predictive power.One resolution to this apparent problem is quite simple; we should un-derstand Einstein gravity as an effective field theory which is a low energyapproximation of an unknown microscopic “UV complete” theory. The ap-propriate microscopic variables may be strings or loops for example. Recallthat higher order interaction terms in the Lagrangian were multiplied bylarger and larger powers of M−1P . At low energies E  MP an nth orderinteraction term Lint ∼ hn is suppressed by a factor (E/MP )n−2. From di-111.3. Feynman-Vernon Influence Functionalmensional grounds the same should be true of the higher order O(Rn) termsparameterized by renormalized coulping constants. The renormalized cou-pling constants should have the form of a dimensionless constant multiplyingan appropriate power of M−1P where the power is determined by the numberof derivatives appearing in the term. Since the derivatives quantify the en-ergy scale of a given process, we see that the higher order terms are supressedby higher and higher order powers of E/MP . Although quantum GR is per-turbatively non-renormalizable, the infinite number of undetermined renor-malized couplings are all coefficients of interaction terms which are highlysuppressed at low energies. There has been significant work done study-ing quantum GR as an effective field theory [68–72]. In principle one canmake predictions accurate to any finite order O ((E/MP )n) and only needto measure a finite number of coupling constants. Since MP ∼ 1018 GeV isan extraordinarily high energy in particle physics contexts these correctionsare indeed very small. Thus as a low energy effective theory perturbativequantum gravity regains its predictive power. This approach is ignorant ofthe details of the underlying high energy theory, which appear only in thevalues of the renormalized couplings. Of course one can immediately seethat at the Planck scale E ∼ MP every interaction term becomes relevantand the theory loses predictive power. This is the regime in which stringtheory or loop quantum gravity may become the appropriate description ofnature.A UV complete theory would be necessary to describe situations in whichthe classical theory predicts extreme curvature, i.e. near black hole or cos-mological singularities. Such extreme scales are however infeasible for man-made experiments. As a result, if QM holds at all scales then the previouslymentioned table-top quantum gravity experiments should be very well de-scribed by the effective field theory of quantum gravity. The lowest ordereffective field theory predictions, involving no additional renormalized cou-plings should be universal and independent of the UV completion of thetheory3.1.3 Feynman-Vernon Influence FunctionalThe primary tool for studying decoherence in the path-integral formalism isthe Feynman-Vernon influence functional (IF) [73]. In this section we willreview the derivation of a general IF. In the next section we’ll discuss why3At this order the cosmological constant is also relevant but we will assume it is negli-gibly small.121.3. Feynman-Vernon Influence Functionalthis simple derivation may be complicated in a gauge theory such as QED orgravity. The primary result of this thesis is a resolution to these difficulties.In what follows we will necessarily use density matrices rather than wave-functions. When describing density matrices ρ(φ;φ′) the number of variablesdoubles but the expressions often contain similar/identical factors for theprimed and unprimed variables. It is convenient to use the following con-densed notation coming from the Schwinger-Keldysh formalism [74, 75]. Fora generic functional f we write f [φ] ≡ f [φ;φ′]. The action will always havethe special form S[φ;φ′] = S[φ]− S[φ′]. For path integrals we write∫ φfφiDφ ≡∫ φfφiDφ∫ φ′fφ′iDφ′. (1.2)Consider a bipartite quantum system. The two interacting quantumsubsystems will referred to as the “central system” and the “environment”.The central system will have states labelled by φ and the environment by X.These are collective variables written in compact notation, both the systemand environment can in principle have many degrees of freedom. The stateof the central system and environment at time ti is described by the densitymatrixρ(φi, Xi). (1.3)Given the initial data, the density matrix at a later time tf ≡ ti + T can bedetermined,ρ(φf, Xf ) =∑φiXiK(φf, Xf ;φi, Xi)ρ(φi, Xi), (1.4)where the density matrix propagator isK(φf, Xf ;φi, Xi) ≡ K(φf , Xf ;φi, Xi)K∗(φ′f , X ′f ;φ′i, X ′i). (1.5)The kernel K is just the usual propagator for the system and environment,i.e.K(φf , Xf ;φi, Xi) = 〈φf , Xf ; tf |φi, Xi; ti〉 (1.6)Supposing only the central system was of observational interest and theenvironment represented some unobserved degrees of freedom, the centralsystem can be fully described by the reduced density matrix obtained bytracing over the environmental degrees of freedomρφ(φf ) ≡∑Xfρ(φf , Xf ;φ′f , Xf ). (1.7)131.3. Feynman-Vernon Influence FunctionalIf the central system and environment are initially uncorrelated the totaldensity matrix factorizes,ρ(φi, Xi) = ρφ(φi)ρX(Xi). (1.8)As time proceeds the interactions between the two will generally lead tocorrelations. The evolution of the system’s reduced density matrix is thengenerally non-unitary since information describing the correlations with theenvironment is lost when the trace is performed. We can describe this non-unitary evolution of the system’s reduced density matrix using an effectivereduced density matrix propagator. The reduced density matrix evolves inthe usual linear wayρφ(φf ) =∑φiKφ(φf ;φi)ρφ(φi), (1.9)and the reduced density matrix propagator isKφ(φf ;φi) ≡∑XfXiδ(Xf −X ′f )K(φf , Xf ;φi, Xi)ρX(Xi). (1.10)Using the path integral representation of the propagator we can proceedto write a path integral representation for the reduced density matrix prop-agator. We will assume the system and environment can be described by anaction of the formS[φ,X] = S[φ] + S[X] + Sint[φ,X]. (1.11)The propagator for the system is thenK(φf , Xf ;φi, Xi) =∫ φfφiDφ eiS[φ]∫ XfXiDX eiS[X]+iSint[φ,X]. (1.12)The reduced density matrix propagator can then be written asKφ(φf ;φi) =∫ φfφiDφ eiS[φ]F [φ], (1.13)where the entire effect of the environment on the system is contained withinthe functional F [φ] called the influence functional [73]. The influence func-tional has the path-integral expressionF [φ] ≡∑XfXiδ(Xf −X ′f ) ρX(Xi)∫ XfXiDX eiS[X]+iSint[φ,X]. (1.14)141.3. Feynman-Vernon Influence FunctionalNote that the double path integral is just a product of propagators for theenvironment subject to a “frozen” background configuration of the centralsystem. Without specifying the system dynamics S[φ] one can still eval-uate the influence functional so long as the interaction Sint[φ,X] and theenvironmental dynamics S[X] are known.A initially thermal environment is commonly used, and is in fact the onewe will use in the next section. It will be useful to have an explicit expressionfor the thermal density matrix. A thermal density matrix corresponding toa canonical ensemble at temperature β−1 has the eigenfunction expansionρβ(X) =∑ne−βEnψn(X)ψ∗n(X′), (1.15)where the ψn(X) are energy eigenfunctions of the Hamiltonian for the envi-ronment alone. Of course this should be divided by the partition functionto ensure the density matrix is properly normalized. To keep the notationcompact we will not explicitly write overall normalizations. Ultimately thenormalization of the influence functional is determined by the fact that itmust equal 1 in the limit that the coupling vanishes. Compare the abovethermal density matrix with the eigenfunction expansion of the propagatorfor the environmentK(X) =∑ne−iEnTψn(X)ψ∗n(X′). (1.16)The thermal density matrix can then be seen as the analytic continuationof the propagator to imaginary times if we make the identification T = −iβ.Analytically continuing the path integral representation of the propagatorto imaginary times we can then write the path integral representation forthe thermal density matrixρβ(X) =∫ XX′DX ′′ e−SE [X′′], (1.17)where SE = −iS|T=−iβ is the Euclidean action. The integration is now overpaths from X ′ → X in imaginary time T = −iβ. The influence functionalfor an initially thermal environment can then be written as the multiplepath integralF [φ] =∑XfXiδ(Xf −X ′f )∫ XfXiDX∫ XiX′iDX ′′eiS[X]+iSint[φ,X]−SE [X′′]. (1.18)151.4. Difficulty Defining an Influence Functional in a Gauge TheoryIn many cases of interest the path integrals cannot be evaluated exactlyand approximate techniques must be used. There is one case of particularrelevance for which the influence functional can be evaluated exactly. Thisis the case that the environment is described by a free action S[X] which isquadratic in X and an interaction which is linear in X, Lint[φ,X] = Xg[φ],for some functional of the system g[φ]. Assuming an initial state whichis gaussian, e.g. a vacuum or thermal state, both the path integrals andthe boundary integrals are gaussian and the influence functional can beevaluated exactly. For a system minimally coupled to a U(1) gauge field,thermal photons provide an environment precisely fitting the above criteria.An analogous computation will show that the same is true in linearizedEinstein gravity.1.4 Difficulty Defining an Influence Functional ina Gauge TheoryEinstein gravity is a beautifully geometric theory. As motivated by theequivalence principle, the theory is coordinate independent. Indeed theEinstein-Hilbert action Eq. (1.1) is invariant under diffeomorphisms4 andthus there is a redundancy in our description of the spacetime manifold.Einstein gravity is then a gauge theory since two metrics which are relatedby a coordinate transformation are considered physically equivalent. In thissection we will discuss how the above discussion of the influence functionalbecomes complicated in gauge theories.The above general discussion of the influence functional required a cleardivide between system φ, and environment X. In gauge theories, this divideis not always straightforward because gauge theories are constrained theo-ries. One says a theory is constrained if its Lagrangian L(φ, φ˙) is singular,i.e. the generalized velocities φ˙a cannot be expressed in terms of the canon-ical momenta pia =∂L(φ,φ˙)∂φ˙a[55]. Classically this implies the existence of aset of primary constraint equations relating the canonical variables,ψm(φa, pia) = 0, (1.19)m = 1, 2, ..., N . The number N of primary constraints depends on the the-ory. Consistent time evolution, ψ˙m = 0, may require that additional “sec-ondary” constraints be appended to the original set of primary constraints.The secondary constraints are treated on the same footing as the primary,4This is only true up to a boundary term which will be further discussed in Section 1.5.161.4. Difficulty Defining an Influence Functional in a Gauge Theoryso this process may be repeated and further tertiary constraints may berequired to ensure the secondary constraints are held throughout time evo-lution. This process is to be repeated until a complete set of consistentconstraints is obtained.The existence of constraints implies that not all of the canonical variablesare independent. Thus in a constrained theory we cannot always naivelypartition the system into a central system and environment where some ofthe generalized coordinates describe the central system and the remainingcoordinates describe the environment. A valid partition must respect theconstraints. A basic example of this is a particle constrained to move ona surface defined by z = f(x, y). Obviously we could not divide the co-ordinates x, y, z of the particle into a central system x, y and environmentz because knowledge of the state of the central system would uniquely de-termine the state of the environment. Even if our measurement apparatuscould only see the shadow of the particle on the x, y plane and not its heightz, we could uniquely determine z. Although the Lagrangian may containthe generalized coordinate variables x, y, z and λ (the Lagrange multiplier)it would be completely incorrect to treat them all as independent quantumoperators unless a restriction is placed on the Hilbert space. Of course thisis a trivial example but it is a useful reminder of this essential idea.In a theory of a gauge field interacting with matter, the constraints aresuch that the state of the gauge field is fundamentally correlated with thestate of the matter. We will see how this comes about shortly. For now wecan see that in order to define an influence functional in a gauge theory onemust first identify the independent variables. Only then can a partition bemade between matter and environment and a partial trace performed.The above considerations were general for constrained theories howevergauge theories are a particular subset of constrained theories. In a gaugetheory there is the additional complication that there is a redundancy in thedescription of the system. The degrees of freedom which are invariant undergauge transformation are called “physical” whereas the remaining degreesof freedom are considered “unphysical”. The two approaches to studyinggauge theories are1. Work in the full (extended) phase space but understand that observ-ables must be gauge invariant quantities.2. Impose a gauge condition which restricts the system to a reduced phasespace in which every degree of freedom is physical.We will choose option 1, and work in the extended phase space with171.5. Quantum Gravity Path Integral - Boundary Termsmanifestly gauge invariant objects. This approach is beneficial because itprevents any doubts about the gauge invariance of the results.1.5 Quantum Gravity Path Integral - BoundaryTerms1.5.1 Gibbons-Hawking-York Boundary TermBefore we can proceed to the quantize the linearized metric perturbationsthere is a technical note which must be made about the Einstein-Hilbertaction Eq. (1.1); the Ricci scalar contains second derivatives of the metric.This causes a problem which may seem purely academic at the classicallevel, but will be very disruptive quantum mechanically.To see how the problem arises classically consider a Lagrangian contain-ing only first derivatives L(φ, ∂µφ). After integration by parts the variationof the action isδS =∫d4x(∂L∂φ− ∂µ ∂L∂(∂µφ))δφ+∮ΣdSµ∂L∂(∂µφ)δφ, (1.20)where Σ is the boundary of the system. The boundary term which is gener-ated by the necessary integration by parts depends linearly on the variationat the boundary. One can assume the variation vanishes along the bound-ary (Dirichlet conditions) and obtain a well defined functional derivativeδS/δφ = (equation of motion). Now consider a Lagrangian depending onsecond derivatives L(φ, ∂2φ). After integration by parts the variation of theaction isδS =∫d4x(∂L∂φ+ ∂2∂L∂(∂2φ))δφ−∮ΣdSµ(∂µ∂L∂(∂2φ)δφ− ∂L∂(∂2φ)∂µδφ).(1.21)Now although the bulk term depends only on the the variation δφ, theboundary term depends both the variation as well as the derivatives of thevariation normal to the boundary surface. To fix both δφ and ∂µδφ equalto zero on the boundary would be equivalent to imposing both Dirichletand Neumann conditions. This cannot be consistently done if the equationof motion is a second order differential equation. Thus the action is notfunctionally differentiable, i.e. δS/δφ 6= (equation of motion). Classicallythis feels like a purely academic problem because the equations of motionare unaffected by the addition of a total derivative term to the Lagrangian.A Lagrangian linear in ∂2φ can be made into a Lagrangian quadratic in ∂µφ181.5. Quantum Gravity Path Integral - Boundary Termsby the addition of a suitable total derivative term. For example, there isusually no concern when one writes the free scalar Lagrangian L = 12φ∂2φsince it is obviously equal to L = −12∂µφ∂µφ up to the addition of a totalderivative 12∂µ(φ∂µφ).At the quantum level, one is not free to simply add total derivatives tothe Lagrangian without affecting the results of the computation. Supposewe had two Lagrangians L1 and L2, where L1 depended on first deriva-tives, L2 on second derivatives, and the two are related by a total derivativeL2 − L1 = ∂µBµ. Classically one would not distinguish between the twosince if one simply ignores boundary terms they produce the same equationof motion. However in quantum mechanics one obtains different results de-pending on which action they choose. Suppose one started with a classicaltheory defined by the Lagrangian L2. Naively quantizing the theory onewould define the propagator asK(φf ;φi) =∫ φfφiDφ ei∫V L2 =∫ φfφiDφ ei∮∂V dSµBµei∫V L1 . (1.22)Although the bulk integrations are unchanged in the choice between L1 andL2, i.e. the stationary phase path is the same in both cases, the phaseaccumulated along the same path will depend on which Lagrangian waschosen.How then are we to decide which of the two Lagrangians is the appro-priate one to use to define the propagator for the quantized theory5? Thisquestion was addressed in quantum gravity by York as well as Hawking andGibbons [76–78]. The linearity of quantum theory implies that propagatorshave the convolution property K(c; a) =∫dbK(c; b)K(b; a). If the actionalong a path between φa and φb is denoted S[b, a], then this convolutionproperty holds if and only ifS[c, a] = S[c, b] + S[b, a]. (1.23)If the action depends on the normal derivative on the boundary then it doesnot satisfy this condition. Taking for example the constant time surfacesta,b,c. If φba(t) is a path between ta and tb, then we can impose the conditionthat both φba(tb) = φb and φcb(tb) = φb, however for general paths the timederivatives will not agree at tb. If the the boundary term in the actiondepended only on the value of φ then it is clear that (1.23) would holdwhereas if the action depended on the time derivative of φ on the boundaries5Furthermore, in general if we have a classical theory defined by a class of equivalentLagrangians, how are we to decide which is the correct one to use in the propagator?191.5. Quantum Gravity Path Integral - Boundary Termsit would not. As a result, in order to maintain the convolution property ofthe propagator we must use classical actions containing only first derivatives.Returning to the problem of interest, since the Ricci scalar contains bothfirst and second derivatives of the metric one must add an appropriate totalderivative to the EH Lagrangian to remove the second derivative terms. Theappropriate term to add is called the Gibbons-Hawking-York (GHY) termSGHY = M2P∮Σd3y√|h|K, (1.24)where Σ is the boundary hypersurface, h is the determinant of the inducedmetric on Σ, K is the trace of the extrinsic curvature Kab of Σ, and  = ±1depending on whether Σ is timelike or spacelike. The correct (containingonly first derivatives) action to include in the path-integral for quantum GRis thusSg = SEH + SGHY =M2P2∫d4x√|g|R+ M2P∮Σd3y√|h|K. (1.25)In the presence of matter one simply adds the covariant matter action6SM =∫d4x√|g| LM (φ, gµν). (1.26)1.5.2 Gauge Transformation Boundary TermWe are interested in making predictions for table-top quantum gravity exper-iments. Such experiments will be well within the regime of weak curvature(the length scales involved are far larger than the Schwarzschild radius ofthe system). In this case we can treat the metric as a perturbation aboutthe Minkowsi metric gµν = ηµν +2MPhµν , where hµν/MP is assumed muchsmaller than 1. The factor of 2 is a matter of preference. We’ll be interestedin the time evolution of the system, which in quantum mechanics corre-sponds to the amplitude to make a transition from an initial state definedon an initial time slice to a final state defined on a later time slice. It is as-sumed that all fields vanish sufficiently fast at spatial infinity so that we canintegrate by parts freely on spatial derivatives without picking up surfaceterms. The relevant boundary Σ = Σi ∪ Σf then consists of two hypersur-faces of constant time. To lowest order in hµν/MP the above action can be6This excludes fermionic matter which couples directly to the connection, as well asnon-minimally coupled matter, e.g. scalar fields coupling directly to the curvature throughterms like φR.201.5. Quantum Gravity Path Integral - Boundary Termswritten as the sum of free terms and an interaction term,S =∮Σd3xhijpi(1)ij −∫ tftid4x(hµνG(1)µν − LM (φ, ηµν)−1MPhµνTµν),(1.27)wherepi(1)ij ≡ K(1)ij − δijK(1) (1.28)is the linearized conjugate momentum to hij ,K(1)ij =12(∂0hij − ∂ih0j − ∂jh0i) (1.29)is the linearized extrinsic curvature, andG(1)µν =12(−∂2hµν − ∂µ∂νh+ ∂ρ∂µhρν + ∂ρ∂νhρµ − ηµν∂σ∂ρhσρ + ηµν∂2h)(1.30)is the linearized Einstein tensor. Indices are now raised and lowered with theMinkowsi metric, and we use the shorthand notation for the trace h = hµµ.The superscript ‘(1)’ is used to emphasize that these quantities are first-order in hµν . In what follows we will drop the superscript ‘(1)’ since allgeometric objects are linearized. The stress-energy tensor Tµν is defined asthe right-hand side of Einstein’s equationTµν = −2∂LM (φ, gµν)∂gµν∣∣∣∣g=η+ ηµνLM . (1.31)It must be mentioned that to the same order in h/MP there is a gravitonself-interaction term of the form hMP (∂h)2. This three-graviton vertex termis the lowest order contribution of the infinitely many non-linear gravitonself-interaction terms. Loosely speaking we can think of (∂h)2 as gravitonstress-energy and the three graviton interaction term as the metric per-turbation coupling to its own stress-energy (as it should according to theequivalence principle). By neglecting this term we are assuming that stress-energy carried by the gravitons is negligible compared to that of the matter.It is not yet clear if this is a valid approximation for the following reasons.This term is responsible for a long-range interaction between gravitons andmatter whereas the hT term which we retain is a local interaction. It is thelong-range interaction which leads to a pole in the forward direction of 2→ 2graviton-matter scattering amplitude analogous to the pole in Rutherfordscattering [79]. Unlike Rutherford scattering where this pole can be ignoreddue to screening effects at long distances, there are no screening effects in211.5. Quantum Gravity Path Integral - Boundary Termsgravity. It will be the topic of future work to assess the validity of ignor-ing the 3-graviton vertex in this context. For now we will operate under thehypothesis that for sufficiently small graviton energies this term is negligible.The full gravitational action is invariant under diffeomorphisms ξ : xµ →ξµ(x), gµν → gξµν which leave the boundaries unchanged [80]. Here gξ isdefined bygµν(x) =∂ξρ(x)∂xµ∂ξσ(x)∂xνgξρσ(ξ(x)). (1.32)This symmetry still holds in the linearized theory so long as hµν/MP  1is preserved. These transformations are of the form xµ → xµ + 2MP ξµ,hµν → hξµν = hµν + ∂µξν + ∂νξµ where ξ is of the same order as hµν . Undera transformation of the above form which does not vanish on the constanttime surfaces the linearized action changes by a boundary termS → S − 2∮Σd3x ξ0(2G00 +M−1P LM). (1.33)Classically these terms are irrelevant but as we’ve previously discussed,boundary terms cannot generally be discarded in quantum theory. Hadwe not included the GHY term in the gravitational action we would nothave obtained this boundary term. We will soon see that this boundaryterm is essential for the implementation of constraints in the path integralformulation of the theory.22Chapter 2Transition Amplitudes inGauge TheoriesAs discussed in the previous section, to define and compute an influencefunctional for a gauge theory we will need to take care of two additionalsubtleties. Firstly, because the theory is constrained we must take caredistinguishing the central system from the environment so that a partialtrace can be performed. Secondly, we must be sure to that the results ofour computations are gauge invariant. In what follows we will see thatboth of these points will be addressed naturally using the path integralrepresentation of the transition kernel.Ultimately we are looking for an effective propagator for the reduceddensity matrix of the central system, and it is clear from equations (1.9)and (1.10) that one first needs the propagator for the joint system. Oneadvantage of the path integral formulation is that it uses the Lagrangianrather than Hamiltonian. In the usual analyses of constrained systems onestarts with a singular Lagrangian, computes the Hamiltonian, determinesthe constraints, and then proceeds to quantize the system. However all ofthe information was contained in the Lagrangian from the start, so thereshould be an equivalent formulation of the quantum theory involving onlythe Lagrangian which still correctly handles the constraints.The first system we will consider is a theory of matter φ, coupled to aU(1) gauge field Aµ. We are using φ as shorthand notation for the mattervariables. If the matter were a complex scalar field φ would represent boththe field and its complex conjugate, if the matter were a Dirac fermion fieldφ would represent a Dirac spinor, etc. The second system we will consider isa theory of matter φ coupled to linearized Einstein gravity (1.27). Althoughwe are ultimately interested in gravity, the calculations will look similar forthe two theories so it will be instructive to study QED first.232.1. Quantum Electrodynamics (QED)2.1 Quantum Electrodynamics (QED)The action for a matter system minimally coupled to a U(1) gauge field canbe written asS[φ,Aµ] =∫ tftid4x(−14FµνFµν +AµJµ + LM), (2.1)where Fµν = ∂µAν − ∂νAµ is the field strength tensor, Jµ is a global U(1)current, and LM depends only on the matter variables. The amplitude forthe system to evolve from a configuration (φi, Aµi (x)) at time ti to configu-ration (φf , Aµf (x)) at time tf ≡ ti + T is given by the propagatorK(Af , φf ;Ai, φi) =∫ φfφ1Dφ eiSM [φ]∫ AfAiDAµ eiS0[A]+i∫ tftid4xAµJµ . (2.2)At first glance it is not clear whether this propagator is even well defined.One possible objection is that in the Hamiltonian formalism one sees thatA0(x) is not a canonical variable and thus does not serve as a label for aquantum state. Furthermore, the above path integral is formally infinite forany choice of boundary data. We will see shortly that both of these issuesare handled naturally in the path integral formalism when gauge invarianceis treated carefully.2.1.1 Free U(1) TheoryBefore we compute the propagator for the full system including matter, wewill familiarize ourselves with the simpler case of free electrodynamics. Inthis case the propagator for the free Maxwell field isK(Af ;Ai) =∫ AfAiDAµ eiS0[A]. (2.3)The propagator is manifestly gauge invariant because the action and measureare gauge invariant. To verify this, consider independent transformations ofthe boundary data, Ai,f → AΛi,f = Ai,f +∂Λi,f . The transformed propagatorisK(AΛff ;AΛii ) =∫ Af+∂ΛfAi+∂ΛiDAµ eiS0[A]. (2.4)242.1. Quantum Electrodynamics (QED)Now simply change integration variables, A′ = A+ ∂Λ, where Λ(x) satisfiesΛ(x, ti,f ) = Λi,f (x). In terms of the primed variable the propagator isK(AΛff ;AΛii ) =∫ AfAiDA′µ eiS0[A′−∂Λ] =∫ AfAiDA′µ eiS0[A′] = K(Af , Ai),(2.5)and the propagator is thus invariant under independent gauge transforma-tions on the boundary data.We can use this gauge invariance to make contact with the canonicalformalism. Following [81] we can perform a spectral decomposition of thepropagator into energy eigenfunctionsK(Af ;Ai) =∑ne−iEnTΨn[Af ]Ψ∗n[Ai]. (2.6)The wavefunctionals Ψ[A] which comprise K will be called physical states.Since the propagator is invariant under independent gauge transformationsof its boundary data, the physical states are gauge invariant Ψ[A] = Ψ[AΛ].We can take this simple equation, and functionally differentiate both sideswith respect to Λ to obtain0 = −i δΨ[AΛ]δΛ(x)∣∣∣∣Λ=0= i∂jδΨ[A]δAj(x)= ∂jEˆj(x)Ψ[A]. (2.7)Thus physical states are wavefunctionals in the extended configuration spacewhich satisfy Gauss’ law as an eigenvalue equation. Actually this equationwas derived by only considering gauge transformations of the spatial com-ponents, in the next section we will proceed more carefully and see that theinvariance of the physical state under timelike gauge transformations impliesthe additional constraint Eˆ0Ψ[A] = 0. To obtain these equations we usedthe operator representation of the canonical momentumpˆiµ =∂ˆL∂(∂0Aµ)≡ −i δδAµ(2.8)and the fact that the electric field Eµ is the (negative of the) conjugate mo-mentum to Aµ. This functional derivative representation can be discussedentirely within the path-integral formalism without passing to a canoni-cal approach (see [82] or Appendix C for details). These physical statesare precisely the states which are considered in Dirac quantization [55, 56].We’ve thus written down the path-integral representation of the propagatorbetween Dirac’s physical states. It is easy to check that this propagator252.1. Quantum Electrodynamics (QED)projects arbitrary sates onto the space of physical states. As a consequence,when evolving an initial state using the transition kernel we can alwaysfirst project the initial state onto the physical subspace. Thus we can workexclusively with physical states without loss of generality.The action is quadratic in the field, so the integral should be naturallyevaluated by shifting variables Aµ = Aˆµ + χµ where Aˆµ is a path which ex-tremizes the action while subject to the boundary conditions Aˆµ(x, ti,f ) =Aµi,f (x). This approach does not work though because there is no uniquesolution to the classical equation of motion subject to these boundary con-ditions. If Aˆµ is a solution then Aˆµ + ∂µΛ is also a solution satisfying theboundary conditions so long as ∂µΛ → 0 as t → ti,f . As a result the inte-gral is infinite. As identified by Faddeev and Popov (FP) this infinite gaugegroup volume can be factored out as a constant overall normalization [83].We will ignore such overall normalizations because the normalization of theinfluence function will be fixed at the end of the computation anyways. TheFP trick is typically used in path integrals which do not have fixed boundarydata. A modification of the FP trick can be used if the boundary data isproperly treated, i.e. integrated over all gauge equivalent configurations, andin fact this procedure implements the constraints which one would find in aHamiltonian framework. This idea was introduced and used in [80, 81, 84]and we will generalize their approach in two ways. Firstly, we generalizefrom their particular gauge choice (temporal gauge) to arbitrary gauge fix-ing functions. Secondly, in the upcoming sections we generalize their resultsto include gauge fields which are coupled to matter.We start by multiplying the propagator by1 =∫DΛ ∆(AΛ)δ(G(AΛ)), (2.9)where Λ(x) is a smooth function vanishing at spatial infinity, G(AΛ) imposesa gauge condition, and ∆(AΛ) = det∣∣∣ δG(AΛ)δΛ ∣∣∣ is the associated FP determi-nant. We can then change variables AΛ = A′ write the path integral asK(Af ;Ai) =∫DΛ∫ AΛffAΛiiDAµ ∆(A)δ(G(A)) eiS0[A]. (2.10)Note that we had to use the gauge invariance of the action and measure toobtain this expression. In the standard application of the FP trick (appliedto integrals without fixed boundary data) the gauge group volume integralfactors out as an overall normalization. In our case the boundary data262.1. Quantum Electrodynamics (QED)ends up depending on the gauge group elements so we cannot immediatelyfactor out this volume. When the propagator is written in this form itsgauge invariance is obvious. The prescription for computing the propagatorinvolves first fixing a gauge and evaluating a gauge dependent integral, thenintegrating the result over all gauge equivalent boundary data.To proceed it is useful to look at the path integral as the limit of discreteintegrals on time slicesK(Af ;Ai) = limN→∞N+1∏n=1∫dΛ(tn)∫ Ajf+∂jΛ(tN )Aji+∂jΛ(t1)dAj(tn) (2.11)×∫ A0f− 1 (Λ(tN+1)−Λ(tN ))A0i− 1 (Λ(t2)−Λ(t1))dA0(tn)∆(A)δ(G(A))eiS[A],where  ≡= tf−tiN . Note that it was necessary to split the vector field intospacelike and timelike components because the gauge transform of the time-like component A0 depends on the time derivative of Λ. In a path integral atime derivative is defined as the difference of a quantity evaluated on consec-utive time slices whereas spatial derivatives are evaluated on a single timeslice. We can now see that the gauge group integrations for n 6= 1, 2, N,N+1can be factored out immediately as an overall normalization since the inte-grand is independent of Λ for intermediate times. The only dependence onΛ(tN+1) and Λ(t2) is in the boundary data for the A0 integral. The inte-grals over Λ(tN+1) and Λ(t2) can then be understood as integrals over theboundary data for A0. The two remaining integrals are over Λ(tN ) ≡ Λfand Λ(t1) ≡ Λi. The propagator can then be written asK(Af ;Ai) =∫dΛfdΛi∫DA0∫ Ajf+∂jΛfAji+∂jΛiDAj ∆(A)δ(G(A))eiS[A]. (2.12)We now see that just as a Hamiltonian formalism would suggest, the propa-gator is independent of A0. It serves as no more than a Lagrange multiplier.We can take this expression further by decomposing the spatial vectorfield into longitudinal and transverse components,ALj =∂j∂k∇2 Ak, (2.13)ATj =(δjk − ∂j∂k∇2)Ak.272.1. Quantum Electrodynamics (QED)We are using the shorthand notation for the Green’s function of the Lapla-cian ∇−2(x). In this notation it is an integral operator,∇−2(x)f(x) = −∫d3yf(y)4pi|x− y| . (2.14)It is obvious that the transverse components are invariant under gauge trans-formation. The integral over gauge equivalent boundary data is thus anintegral over the longitudinal part of the boundary data. We then arrive atthe expression for the propagatorK(Af ;Ai) =∫DA0DAL∫ ATfATiDAT ∆(A)δ(G(A))eiS[A]. (2.15)This is precisely the Faddeev formula applied to free U(1) gauge theory[85]. Faddeev derived his formula however by passing first to a canonicalHamiltonian framework and then later constructing the path integral. In ourderivation the Lagrangian was used start to finish and the physical degreesof freedom emerged naturally when we integrated over gauge equivalentboundary data. For the specific choice of temporal gauge G(A) = A0 thisformula was obtained in a similar manner by [81].It is convenient to choose the Coulomb gauge G(A) = ∂jAj . In this caseboth the timelike and longitudinal integrals can be done and their resultsalong with the FP determinant can be factored out as overall constants.The kernel then takes the simple form in terms of only the transverse fieldcomponents (i.e. the radiation),K(Af ;Ai) =∫ ATfATiDAT eiS[AT ], (2.16)where the action for the transverse components isS[AT ] = −12∫ tftid4x(∂µATj )(∂µAj T ) = −12∫ tftid4xPµν∂σAµ∂σAν . (2.17)This is written in terms of the transverse projector Pµν =∑a µaνa, herewritten in terms of the two orthogonal transverse polarization vectors. Ex-plicitly, P 0µ = 0, P ij =(δij − ∂i∂j∇2). Clearly this is not Lorentz invariantbut that is expected since Lorentz invariance was broken when we identifiedpreferred spacelike slices Σi,f .The remaining integral is easily evaluated by shifting the integrationvariable by the classical solution. The fluctuation determinant factors out282.1. Quantum Electrodynamics (QED)as an overall constant and we’re left with the result given in terms of theclassical action for the transverse field evolving between ATi and ATf in timeT = tf − ti,K(Af ;Ai) = eiScl[ATi →ATf ]. (2.18)The expression for the classical action is presented in Appendix A.Regardless of the longitudinal or timelike data we fix at the endpoints,the gauge invariance of the action has made the propagator depend onlyon the boundary data which lies in the gauge invariant subspace of the fullconfiguration space (i.e. the transverse components).2.1.2 Interacting QEDNow that we have seen the simple example of free U(1) theory let’s seehow the computation of the propagator is changed when we include chargedmatter. Of course we cannot perform the integral over the gauge field and thematter variables but for the purposes of computing an influence functionalwe only need to perform the gauge field integral. The QED propagator isK(Af , φf ;Ai, φi) =∫ φfφiDφ∫ AfAiDAµ eiS[A,φ]. (2.19)The action is invariant under the U(1) gauge transformation φ→ φΛ = eiΛφ,Aµ → AΛµ = Aµ + ∂µΛ. As a result the propagator is invariant underindependent gauge transformations of the boundary data,∫ φΛffφΛiiDφ∫ AΛffAΛiiDAµ eiS[A,φ] =∫ φfφiDφ′∫ AfAiDA′µ eiS[(A′)−Λ,(φ′)−Λ] (2.20)=∫ φfφiDφ∫ AfAiDAµ eiS[A,φ].The first equality was merely a change of variables, φ′ = φΛ,A′µ = AΛµ ,where the gauge transformation Λ(x) matches the gauge transformations onthe boundaries, Λ(x, ti,f ) = Λi,f (x). The second equality used the gaugeinvariance of the action. Thus the propagator is indeed invariant underindependent gauge transformations of its boundary data,K(AΛff , φΛff ;AΛii , φΛii ) = K(Af , φf ;Ai, φi). (2.21)This invariance implies that the physical states which comprise the prop-agator are also gauge invariant,Ψ[AΛ, φΛ] = Ψ[A, φ]. (2.22)292.1. Quantum Electrodynamics (QED)This expression can be written in a more illuminating way if we use thefollowing functional identities,f [φ+ g] = e∫d3xg δδφ f [φ], (2.23)f [eiΛφ] = ei∫d3xΛφ δδφ f [φ]. (2.24)The first identity is a straightforward application of a linear shift operator,while the second is a slight generalization from the group of translations tothe group of dilatations [86]. The second identity can be rewritten by notingthat the U(1) Noether charge is defined asJ 0 = −iφ ∂L∂(∂0φ)= −iφΠ, (2.25)where Π is the conjugate momentum to the field φ. Writing the conjugatemomentum in its operator representation we can then see that the U(1)transformation on a functional is generated by the charge densityf [eiΛφ] = e−i∫d3xΛJˆ 0f [φ]. (2.26)With these identities we can write Eq. (2.22) asexp[∫d3x ∂tΛδδA0+ Λ(−∂j δδAj+ iφδδφ)]Ψ[A, φ] = Ψ[A, φ]. (2.27)or equivalently[∫d3x ∂tΛδδA0+ Λ(−∂j δδAj+ iφδδφ)]Ψ[A, φ] = 0 (2.28)Since ∂tΛ is a variable independent of Λ and this equation holds for arbitraryvalues of these parameters, the coefficient of ∂tΛ and the coefficient of Λmust vanish independently. Rewriting the functional derivatives in terms asthe operators which they represent we obtain two equations which physicalstates satisfyEˆ0Ψ[A, φ] = 0 (2.29)and (∂jEˆj − Jˆ 0)Ψ[A, φ] = 0. (2.30)These are precisely the constraint equations one would impose in the canoni-cal Dirac quantization of this system. As a consequence of gauge invariance,physical quantum states still satisfy Gauss’ law as an eigenvalue equation.302.1. Quantum Electrodynamics (QED)As in free U(1) theory, the path integral is formally divergent and againthe gauge group volume must be factored out using the modified FP trick.Multiplying the above equation by (2.9) and changing variables as we didpreviously we obtainK(Af , φf ;Ai, φi) =∫DΛ∫ φΛffφΛiiDφ∫ AΛffAΛiiDAµ ∆(A)δ(G(A))eiS[A,φ].(2.31)The gauge transformation on the matter boundary data can be rewritten inan illuminating way if we use identity (2.24). The propagator can then bewritten asK(Af , φf ;Ai, φi) = (2.32)=∫DΛ UˆΛ(tf )∫ φfφiDφ∫ AΛffAΛiiDAµ ∆(A)δ(G(A))eiS[A,φ] Uˆ †Λ(ti),where the operator which effects gauge transformations on the matter vari-ables isUˆΛ(t) = e−i ∫ d3xΛ(x,t)Jˆ 0(x,t). (2.33)As we did in the above example of free U(1) theory, we will re-express thispath integral as the limit of discrete integrals on constant time slices,K(Af , φf ;Ai, φi) = limN→∞N+1∏n=1∫dΛ(tn) (2.34)× UˆΛ(tN )∫ φfφidφ(tn)∫ AΛffAΛiidAµ(tn) ∆(A)δ(G(A))eiS[A,φ] Uˆ †Λ(t1).Again, the gauge transformations of the boundary data are of the formAµi,f + ∂µΛi,f . The spatial gradients are defined on a single time-slice, butthe time derivatives are are a difference between variables on consecutivetime-slices, ∂0Λ(t1) =1 (Λ(t2)−Λ(t1)), and ∂0Λ(tN ) = 1 (Λ(tN+1)−Λ(tN )).Since the integral depends only on Λ(tn) for n = 1, 2, N,N + 1 the gaugegroup volume can be factored out for intermediate times. As well since theonly dependence on Λ(t2,N+1) is in the gauge transform of A0i,f , the integralsover Λ(t2,N+1) act as integrals over the boundary data for A0. Only two312.1. Quantum Electrodynamics (QED)integrals remain, and the propagator can be written asK(Af , φf ;Ai, φi) =∫dΛidΛf (2.35)× UˆΛ(tf )(∫ φfφiDφ∫DA0∫ Ajf+∂jΛfAji+∂jΛiDAj ∆(A)δ(G(A))eiS[A,φ])Uˆ †Λ(ti).The simplest choice of gauge fixing function is Coulomb gauge G(A) = ∂jAj .In this case the FP determinant factors out as an overall constant. Forintermediate times this gauge choice fixes the longitudinal part of the field tobe zero. The gauge fixing delta functions for the boundary times δ(G(Ai,f ))then impose the condition ∂jAji,f = −∇2Λi,f which can be solved for Λi,f .The integrals over Λi,f can then be performed using the Coulomb gaugefixing delta functions and the resulting propagator isK(Af , φf ;Ai, φi) = UˆC(tf )(∫ φfφiDφ∫DA0∫ ATfATiDAT eiS[A0,AT ,φ])Uˆ †C(ti).(2.36)Here we’ve introduced the notationUˆC(t) = exp(−i∫d3xAj(x, t)Cˆj(x, t)), (2.37)where Cˆj(x, t) = − ∂∂xj∫d3y 14pi|x−y| Jˆ 0(y, t) is the Coulomb electric field cre-ated by charge density Jˆ 0. The operator UˆC acts in a very intuitive way.If we consider a state Ψ[φ] describing the matter then the state UˆCΨ[φ]satisfiesEˆj(x)UˆCΨ[φ] = iδδAj(x)exp(−i∫d3yAj(y, t)Cˆj(y, t))Ψ[φ] (2.38)=(− ∂∂xj∫d3yJˆ 0(y)4pi|x− y|)UˆCΨ[φ].Thus the action of the operator UˆC is to create a coherent Coulomb electricfield around the matter. The Coulomb electric field is the solution to theGauss law constraint equation, so UˆC makes a state satisfy Gauss’ law andthus be gauge-invariant. As a result of Eqs. (2.29) and (2.30) physical statesare of the formΨ[A, φ] = UˆC ψ[AT , φ]. (2.39)322.1. Quantum Electrodynamics (QED)Physical states consist of transverse photons and matter with its accompa-nying Coulomb field. All of the dependence on the longitudinal part of thefield is in UˆC and the state does not depend on A0. The unconstrained partof the wavefunctional ψ[AT , φ] can take on any form.This is reminiscent of Dirac’s approach to gauge invariant QED [87].He constructed gauge invariant fermion field operators which created ana electron with an accompanying coherent gauge field. In his case it wasambiguous what the accompanying field should be since any transverse fieldcould be added to the Coulomb field and the resulting field operator wouldstill be gauge invariant. We do not have this ambiguity since all of thedependence on the transverse field lies in ψ[AT , φ]. It should be emphasizedthat our results are indeed different. We are describing general dressed statesnot dressed field operators.Returning to the evaluation of the path integral, in Coulomb gauge theaction isS[A0, AT , φ] = SM [φ]+Sγ [AT , φ]+∫ tftid4x(12A0∂j∂jA0 +A0J0), (2.40)where the action for transverse photons isSγ [AT , φ] =∫ tftid4xPµν(−12∂σAµ∂σAν +AµJν). (2.41)The A0 integral can be immediately done and it merely adds a new interac-tion term to the matter action corresponding to the Coulomb force betweenthe charge densities,∫DA0 eiS[A0,AT ,φ] = eiS[AT ,φ]+iSC [φ] (2.42)where the instantaneous Coulomb interaction term isSC [φ] = −12∫ tftidt∫d3xd3yJ0(x, t)J0(y, t)4pi|x− y| . (2.43)With the timelike and longitudinal integrals done the final expression forthe propagator is written in terms of only the independent dataK(Af , φf ;Ai, φi) = UˆC(tf )(∫ φfφiDφ∫ ATfATiDAT eiS[AT ,φ]+iSC [φ])Uˆ †C(ti),(2.44)332.2. Linearized Quantum Gravityor equivalentlyK(Af , φf ;Ai, φi) = UˆC(tf )(∫ φfφiDφ eiSM [φ]+iSC [φ])Uˆ †C(ti) (2.45)×∫ ATfATiDAT eiSγ [AT ,φ].The latter form is convenient because the remaining path integral on thegauge field has been separated from the matter integral and can be evaluatedas the integral for a transverse electromagnetic radiation field on a frozenbackground source field.The path integral for the transverse field can again be evaluated by sim-ply shifting the integration variable by the classical solution. The fluctuationdeterminant again factors out as an overall constant and the result is justthe classical action for the transverse field on a frozen background sourcefield,K(Af , φf ;Ai, φi) = UˆC(tf )(∫ φfφiDφ eiSM [φ]+iSC [φ])Uˆ †C(ti) eiScl[ATi →ATf ,φ],(2.46)where Scl[ATi → ATf , φ] is a straightforward to compute but lengthy expres-sion which we present in Appendix A.As a result of gauge invariance there are constraints implemented onthe system enforcing that i) the kernel is independent of A0 and ii) thedependence of the kernel on AL is determined entirely by the charge densityof the matter (Gauss’ law). This implies that physical states are of the formUˆC ψ[AT , φ]. Indeed the only components of the electromagnetic field whichare independent of the matter are the transverse components. ComparingEq. (2.46) to Eq. (1.12) we see that the natural partition into central systemand environment is not matter and gauge field, but rather matter (with itscoherent Coulomb field) and radiation field. A partial trace which resultsin a physical state must then be only over the transverse photon degrees offreedom since the longitudinal degrees of freedom are constrained by Gauss’law. This will be used in Section 3 to compute the influence functional.2.2 Linearized Quantum GravityNow that we have familiarized ourselves with the computation of a gaugeinvariant propagation kernel in a simple gauge theory lets proceed to quan-tum gravity. As mentioned in the introduction we will use the effective field342.2. Linearized Quantum Gravitytheory approach to quantum gravity in the low energy linearized approxi-mation.The action for matter coupled to linearized Einstein gravity (includingthe GHY term) is given in Eq. (1.27),S[h, φ] =∮Σd3xhijpi(1)ij −∫ tftid4x(hµνG(1)µν − LM (φ, ηµν)−1MPhµνTµν).(2.47)The amplitude to evolve from an initial configuration (φi, hµνi ) to a laterconfiguration (φf , hµνf ) is given by the path integral representation of thepropagatorK(hf , φf ;hi, φi) =∫ φfφiDφ∫ hfhiDhµν eiS[h,φ]. (2.48)Again, as with Eq. (2.2) one may take objection to this expression since weknow from the canonical formalism that h0ν are not canonical variables, aswell as the path integral being formally divergent. We’ve seen already inSections 2.1.1 and 2.1.2 that these issues were resolved naturally by carefullyfactoring out the gauge redundancy from the integral. The same is true ofthis gravitational path integral.As emphasized in Section 1.5.2 this action is not invariant under gaugetransformations but rather it changes by a boundary term (1.33). Letssee what this implies for the propagator. A gauge transformation of theboundary data causes a change in the propagatorK(hξff , φξff ;hξii , φξii ) = (2.49)= e−2i ∫Σf ξ0(2Gˆ00+ 1MP LˆM)K(hf , φf ;hi, φi)e2i ∫Σi ξ0(2Gˆ00+ 1MPLˆM),where φξ denotes the matter variables under the transformation xµ →xµ + 2MP ξµ. The propagator is then invariant under (small) spatial dif-feomorphisms but not under diffeomorphisms changing the initial and finaltime coordinates. The physical states which comprise the propagator thentransform asΨ[hξ, φξ] = e−2i ∫ d3x ξ0(2Gˆ00+ 1MP LˆM)Ψ[h, φ]. (2.50)This expression can be rewritten in a more physically illuminating way.Since ξ/MP  1 we can write a functional f [φ] of the transformed variableusing a linear shift operatorf [φξ] = f[φ+2MPξµ∂µφ]= e2MP∫d3x ξµ(∂µφ)δδφ f [φ]. (2.51)352.2. Linearized Quantum GravityThe metric perturbation also transforms by a linear shift, so we can writef [hξµν ] = e2∫d3x(∂µξν)δδhµν f [hµν ]. (2.52)We now rewrite the functional derivatives in terms of the operators whichthey represent, −i δδφ = Πˆ and −i δδhµν = pˆiµν where Π =∂LM∂(∂0φ)and piµν =∂Lg∂(∂0hµν). The definition of the stress-tensor as the Noether current associatedwith spacetime translations isTµν = −(∂νφ) ∂LM∂(∂µφ)+ ηµνLM , (2.53)so as an operator the 4-momentum density can be writtenTˆ 0ν = i(∂νφ)δδφ+ η0νLˆM . (2.54)The functional identities are then expressible as,f [φξ] = e−i 2MP∫d3x ξµ(Tˆµ0−ηµ0LˆM)f [φ], (2.55)f [hξµν ] = e2i∫d3x (∂νξµ)pˆiµν (2.56)Another useful formula is the relation between the conjugate momentumto the metric perturbation and the Einstein tensor, −∂jpijk = 2G0k. Usingthese relations we can rewrite Eq. (2.50) asexp[2i∫d3x((∂0ξν)pˆi0ν + ξν(2Gˆ0ν −M−1P Tˆ 0ν))]Ψ[h, φ] = Ψ[h, φ],(2.57)or equivalently[∫d3x((∂0ξν)pˆi0ν + ξν(2Gˆ0ν −M−1P Tˆ 0ν))]Ψ[h, φ] = 0 (2.58)Since this equation holds for arbitrary ξµ and ∂0ξµ, and they are independentparameters, their coefficients must independently vanish. We then obtaintwo constraint equations which the physical states satisfy,pˆi0νΨ[h, φ] = 0, (2.59)(Gˆ0ν − 12MPTˆ 0ν)Ψ[h, φ] = 0. (2.60)362.2. Linearized Quantum GravityThe first equation says that physical states do not depend on the time-like components h0ν . The second equation can be identified as the (lin-earized) Hamiltonian and momentum constraint equations of general rela-tivity. These are precisely the constraint equations one would impose onphysical states in the Dirac canonical formalism [55, 56]. Just as we didin QED, here we obtain these constraint equations as a consequence of thegauge invariance of the propagator.Now that we’ve seen how the gauge invariance of the propagator impliesthat physical states satisfy certain constraint equations, we can proceed toevaluate the path integral (for the metric variables). This proceeds com-pletely analogously to the QED example. To factor out the gauge redun-dancy we multiply by1 =∫Dξ∆(hξ)δ(G(hξ)). (2.61)Note that the integral is over ξµ including those which change the bound-aries. We multiply by the above FP factor and change variables to obtainK(hf , φf ;hi, φi) =∫Dξµ∫ φξffφξiiDφ∫ hξffhξiiDhµν (2.62)×∆(h)δ(G(h)) eiS[h,φ]+2i∮Σ d3x ξ0(2G00+M−1P LM).Using the identities Eq. (2.55) and Eq. (2.56) we can rewrite the propagatorin the convenient formK(hf , φf ;hi, φi) = (2.63)=∫Dξµ Uˆξµ(tf )∫ φfφiDφ∫ hξffhξiiDhµν ∆(h)δ(G(h)) eiS[h,φ] Uˆ †ξµ(ti),where we’ve defined the operatorUˆξµ(t) = e2i∫d3x ξ0(2G00−M−1P Tˆ 00)−M−1P ξj Tˆ 0j . (2.64)The next step is to cut the path integral into discrete time slices. Recallhow the gauge transform of A0 depended on the time derivative ∂0Λ andthis allowed us to use rewrite the integrals∫dΛ(t2)∫dΛ(tN+1) as integralsover the boundary data for A0. The same situation occurs here for h0ν . Thegauge transform of the spatial components hξjk depends only only spatial372.2. Linearized Quantum Gravityderivatives and thus only on ξj at times t1,N ≡ ti,f . The gauge transform ofthe timelike components ishξ0ν(tf ) = h0µ(tf ) + ∂νξ0(tN ) +1(ξν(tN+1)− ξν(tN )) (2.65)hξ0ν(ti) = h0µ(ti) + ∂νξ0(t1) +1(ξν(t2)− ξν(t1))A simple change of variables allows us to write the integrals over ξν(tN+1)and ξν(t2) as integrals over the boundary data for h0ν . Once this is donethe boundary data is independent of ξ0. In fact the only dependence onξ0 is in the boundary operator (2.64). The integrand in Eq. (2.63) doesn’tdepend on ξµ for intermediate times and thus the gauge group volume canbe factored out as an overall normalization. The resulting path integral isK(hf , φf ;hi, φi) =∫dξjf∫dξji (2.66)× δ(Hˆ)Uˆξj∫ φfφiDφ∫Dh0ν∫ hξffhξiiDhjk ∆(h)δ(G(h)) eiS[h,φ] Uˆ †ξjδ†(Hˆ).The operatorδ(Hˆ) ≡∫dξ0 e4i∫d3x ξ0(G00− 12MPTˆ 00)(2.67)is the projector onto the kernel of the operator Gˆ00 − 12MP Tˆ 00. That is,it projects onto the subspace of the Hilbert space satisfying the Hamilto-nian constraint HˆΨ = 0. Since the overall normalization is irrelevant, wecan assume that δ(Hˆ) returns 1 when acting on a state which satisfies theHamiltonian constraint and returns zero otherwise.To evaluate the path integral it is convenient to further decompose themetric perturbation. Similar to the transverse and longitudinal decompo-sition of a vector field, a symmetric tensor field can be decomposed intolongitudinal, transverse-trace, and transverse-traceless (TT) partshjk = hLjk + hTjk + hTTjk (2.68)which satisfy ∂jhTjk = 0, ∂jhTTjk = 0, and δjkhTTjk = 0. Explicit expressionsare obtained using the transverse projector Pij ,hLjk =(δaj δbk − P aj P bk)hab, (2.69)hTjk =12PjkPabhab, (2.70)382.2. Linearized Quantum GravityhTTjk =(P aj Pbk −12PjkPab)hab. (2.71)It is easy to check that only the longitudinal part transforms under gaugetransformation.Along with this decomposition we must choose a gauge fixing function.The most convenient choice is transverse gauge G(h) = ∂jhjν = 0. With thischoice the FP determinant factors out as an overall constant. This gaugechoice sets the longitudinal part of the field to zero for intermediate times.For times ti,f it enforces ∂jhLjk +∇2ξk + ∂j∂kξj = 0. This can be solved forξj(ti,f ) to findξj(ti,f ) = − 1∇2(δja −∂j∂a2∇2)∂bhabi,f . (2.72)We are now able to evaluate the integrals over all of the different componentsof the metric perturbation h00, h0j , hTjk, hLjk, and hTTjk . The h00 integral canbe done immediately since h00 appears only linearly in the action. It simplyproduces a delta function enforcing the Hamiltonian constraint for everyintermediate time in the path integral. The Hamiltonian constraint fixes thetransverse-trace part of the metric perturbation in terms of the longitudinalpart and the matter stress tensor. Using the Hamiltonian constraint thetransverse-trace part can then be immediately integrated. Next the h0jintegral can be performed and it is a straightforward gaussian integral. Theresult is an instantaneous gravitational interaction term added to the matteraction analogous to the A0 integration in QED generating the Coulombinteraction term. The last remaining integral is the trivial integral is overthe longitudinal part.The resulting path integral expression for the gravitational propagatorisK(hf , φf ;hi, φi) =δ(Hˆ) UˆG(∫ φfφiDφ eiSM [φ]+iSSG[φ])Uˆ †Gδ†(Hˆ) (2.73)×∫ hTTfhTTiDhTTjk eiSg [hTT ,φ].We have defined the instantaneous gravitational self-interaction termSSG[φ] = − 14M2P∫ tftid4x (2.74)× 1∇2(T 00T 00 − 4T 0jPjkT 0k + 2T 00PjkT jk + ∂0T00∂0T00∇2).392.2. Linearized Quantum GravityThe first term is simply the Newton potential while the rest of the termsare purely relativistic. This expression may seem unfamiliar but it is indeedthe correct gravitational analogue of the Coulomb interaction in QED. Thisis demonstrated in Appendix B. The transverse-traceless action isSg[hTT , φ] =∫ tftid4xΠµναβ(−12∂σhµν∂σhαβ +1MPhµνTαβ), (2.75)written in terms of the TT projector Πµναβ = 12(PµαP νβ + PµβP να − PµνPαβ).The operators UˆG are defined asUˆG = exp(i1MP∫d3xhjkBˆjk), (2.76)whereBˆjk = − 1∇2(δjl∂k + δkl∂j − ∂j∂k∂l∇2)Tˆ 0l. (2.77)To see how UˆG acts consider a state Ψ[φ] describing only the matter.The state UˆGΨ[φ] then satisfies∂j(iδδhjk)UˆGΨ[φ] = − 1MP∂jBˆjkUˆGΨ[φ] = Tˆ0kUˆGΨ[φ]. (2.78)This can be rewritten in terms of the conjugate momentum pˆijk and we cansee that the state UˆGΨ[φ] satisfies the momentum constraintPˆkUˆGΨ[φ] ≡(Gˆ0k − 12MPT 0k)UˆGΨ[φ] = 0. (2.79)In the same way the operator UˆC created a coherent Coulomb electric fieldin QED (ensuring the that physical states satisfy Gauss’ law), the operatorUˆG creates a coherent gravitational field which ensures that physical statessatisfy the momentum constraint and are thus invariant under spatial gaugetransformations. As a result of the constraints Eq. (2.59) and Eq. (2.60),physical states are of the form Ψ[h, φ] = δ(Hˆ) UˆG ψ[hTT , φ]. They are inde-pedent of the timelike components h0ν , and the dependence on the longitu-dinal field is entirely in UˆG. The unconstrained part of the wavefunctionalψ[hTT , φ] can take on any form, since hTT and φ are the true independentdegrees of freedom. The only dependence on the transverse-trace part of thefield is through the projector δ(Hˆ). This projector ensures physical wave-functions are non-zero only if the transverse-trace part of the field takes402.2. Linearized Quantum Gravityon a value such that the Hamiltonian constraint is satisfied. This is quiteunlike the momentum constraint which is satisfied by dressing matter stateswith coherent fields using the operator UˆG. The reason for this differenceis that the momentum constraint is a relation between the coordinates andmomenta (hjk, pijk) whereas the Hamiltonian constraint is a constraint onlyon the coordinates, G00 = 12(∂j∂k − δjk∇2)hjk. The Hamiltonian constraintthen restricts physical states to be wavefunctionals which only have supporton the kernel of Hˆ rather than wavefunctionals with support on the entirespace of three-metric configurations. This is a point argued by Kuchar [88]which was imposed by hand in [89] and derived in a way very similar to usin [80].Finally, the path integral for the TT parts of the metric perturbation canbe done by shifting the integration variable by the classical solution. Thefluctuation determinant factors out as an overall constant and the result isthe classical action for the TT field on a frozen background source. Thefinal expression for the propagator, having evaluated the gravitational pathintegral isK(hf , φf ;hi, φi) = δ(Hˆ) UˆG(∫ φfφiDφ eiSM [φ]+iSSG[φ])Uˆ †G δ†(Hˆ) eiScl[hTTi→f ,φ](2.80)The classical action Scl[hTTi→f , φ] is presented in Appendix A.As a result of gauge invariance, there are constraints implemented on thesystem enforcing that i) the propagator is independent of h0ν ii) the depen-dence of the propagator on hL is determined by the momentum density ofthe matter (due to the momentum constraint) and iii) the dependence of thepropagator on the trace δjkhjk is constrained by energy density of the mat-ter (due to the Hamiltonian constraint). As a result the only independentdegrees of freedom are the matter degrees of freedom and the transverse-traceless graviton degrees of freedom. Physical states can then be written asΨ[h, φ] = δ(Hˆ) UˆG ψ[hTT , φ]. The appropriate partition into central systemand environment is then matter (dressed by appropriate coherent gravita-tional field) and transverse-traceless gravitons. A partial trace resulting ina state which is still physical must then be only over the transverse-tracelessgraviton degrees of freedom.We’ve now seen three examples of how the gauge invariance of the prop-agator leads to the idea of gauge invariant physical states. These physicalstates which comprise the propagator are annihilated by operators whichgenerate gauge transformation. Such equations are precisely the constraintequations which one imposes in Dirac’s canonical quantization of constrained412.2. Linearized Quantum Gravitysystems. We computed the path integrals for the gauge fields using a modi-fication of the FP trick. The result is a manifestly gauge invariant propaga-tor which is equivalent to the propagator one would obtain in the canonicalDirac formalism (see [53]). As a result of our computations it is clear how todivide the central system from the environment. In QED the environmentis the transverse (photon) part of the gauge field, while the rest of the gaugefield is constrained by the state of the matter. In linear quantum gravitythe environment is the transverse-traceless (graviton) part of the gauge field,while the rest of the gauge field is constrained by the state of the matter.The physical picture in QED is that of dressed charges (charges surroundedby their associated Coulomb electric field) interacting with a field of trans-verse photons. The picture in gravity is similar, there is matter dressed bya gravitational field and the matter interacts with TT gravitons. In thenext chapter we will take these results and compute the influence functionaldescribing the interaction of these central systems with the radiation envi-ronment in each theory.42Chapter 3Influence Functional in QEDand linearized Gravity3.1 Thermal Photon Bath Influence FunctionalIn section 2.1.2 we showed that as a result of gauge invariance the propa-gator is composed of physical states Ψ[A, φ] which have the form Ψ[A, φ] =UˆC ψ[AT , φ], which makes it clear that the only independent degrees of free-dom are that of the matter and transverse field components. It was alsomentioned that because the propagator projects onto the space of physicalstates, we can work exclusively with physical states without loss of general-ity. Physical density matrices are of the formρ[A, φ] = UˆC ρ˜[AT , φ] Uˆ †C . (3.1)We will use tilde to denote density matrices which only depend on matterand/or transverse field components. The tilded density matrices describeunphysical states but the appropriate physical state can be obtained simplyby conjugating ρ˜ with the operator UˆC . For brevity we can work with thedensity matrices ρ˜.The transverse field components can be traced out to obtain the reduceddensity matrix for the matterρ˜φ[φ] =∫dAT δ(AT −AT ′)ρ˜[AT , φ]. (3.2)For physical states it is possible that the transverse degrees of freedom areuncorrelated with the matter, but obviously the longitudinal part of the fieldmust be correlated with the matter. A state in which the transverse fieldcomponents are uncorrelated with the matter is written as a product stateρ˜[AT , φ] = ργ [AT ]ρ˜φ[φ].Now we can return ourselves to the overarching question related to de-coherence. Suppose the environment, i.e the transverse field components, isinitially uncorrelated with the matter. We can evolve the state forward in433.1. Thermal Photon Bath Influence Functionaltime and trace out the environment to obtain the final (physical) state ofthe matter. The reduced density matrix for the matter can then be writtenasρφ[φf ] = UˆC(∫dφiρ˜φ[φi]∫ φfφiDφ eiSM [φ]+iSC [φ]F [φ])Uˆ †C , (3.3)and the influence functional is given byF [φ] =∫dATf∫dATi δ(ATf −AT ′f )ρ[ATi ]∫ ATfATiDAT eiS[AT ,φ]. (3.4)Equations (3.3) and (3.4) are of precisely the same form as Eqs. (1.13)and (1.14) for an environment consisting of transverse photons. The upshotof this manifestly gauge invariant computation is that it has become clearthat only the transverse photons can act as an environment in QED whilethe timelike and longitudinal parts of the gauge field act to produce theCoulomb interaction in the matter action and to dress the matter with acoherent Coulomb electric field.At this point it is straightforward to compute the influence functionalfor an initially thermal photon bath. A state describing a thermal gaugefield is of the formρ˜[AT , φ] = ρ˜φ[φ]∫ ATAT ′DAT ′′ e−SE [AT ′′]. (3.5)The influence functional for a bath of thermal photons is then given by thefunctional integralF [φ] =∫dATf δ(ATf −AT ′f )∫dATi∫ ATfATiDAT∫ ATiAT ′iDAT ′′ eiS[AT ,φ]−SE [AT ′′].(3.6)Each of the functional integrals is gaussian and can be evaluated immediatelyby shifting the integration about the classical path. The remaining integralsare gaussian as well and are done by direct integration. At this stage theenvironment has been reduced to a collection of independent oscillators,a well studied case [73, 90]. For a theory in which the U(1) current isindependent of Aµ (this excludes scalar QED for example) and the couplingterm is Lint = eAµJ µ the result is expressed conveniently in terms of aninfluence phase,F [φ] = eiΦ[φ]. (3.7)443.2. Thermal Graviton Bath Influence FunctionalThe influence phase is given by the expressioniΦ[φ] =i∫ tftid4x∫ x0tid4x˜[Jµ(x)− J ′µ(x)] (3.8)×([Jν(x˜) + J ′ν(x˜)] γµν(x− x˜) + i [Jν(x˜)− J ′ν(x˜)] ηµν(x− x˜)).The dissipation and noise kernels are respectively given byγµν(x) =e22∫d3p(2pi)3eip·x Pµν(p)sin px0p, (3.9)ηµν(x) =e22∫d3p(2pi)3eip·x coth(βp2)Pµν(p)cos px0p, (3.10)where β is the inverse temperature.We can now finally write down the physical (gauge-invariant) reduceddensity matrix for matter interacting with a thermal bath of photons,ρφ[ALf , φf ] =∫dφiρ˜φ[φi](UˆC∫ φfφiDφ eiSM [φ]+iSC [φ]+iΦ[φ] Uˆ †C). (3.11)It should be noted that the influence functional Eq. (3.8) is not a newresult, see [91]. The manifestly gauge invariant computation is new how-ever. Regardless, the QED computation was considered a warm-up for thequantum gravity computation which has had more debate in the literature.3.2 Thermal Graviton Bath Influence FunctionalThe discussion for linear gravity is nearly identical to the above for QED.We’ve seen in section 2.2 that as result of gauge invariance physical statesare of the form Ψ[h, φ] = δ(Hˆ)UˆGψ[hTT , φ]. The only independent degreesof freedom are the TT gravitons and the matter. We can write physicaldensity matrices in the formρ[h, φ] = δ(Hˆ) UˆG ρ˜[hTT , φ] Uˆ †G δ†(Hˆ). (3.12)All of the dynamics is described by the evolution of the unphysical tildeddensity matrix, and the physical state can be obtained at the end of thecomputation by replacing the operators δ(Hˆ) UˆG.453.2. Thermal Graviton Bath Influence FunctionalA state in which the environment (TT gravitons) is initially uncorrelatedwith matter as well as being thermal can be written asρ˜[hTT , φ] = ρ˜φ[φ]∫ hTThTT ′DhTT ′′ e−SE [hTT ′′]. (3.13)Assuming an initial condition of this form is convenient from a calculationalperspective since it allows for an exact evaluation of the influence functionalintegrals, but we must ask if it is physically reasonable. As we’ve previouslyemphasized, gravity is unique because it cannot be screened. This impliesthat the central system and graviton environment will always be correlatedto some degree. In our computation we will assume the above simple formfor the initial state, but this must be understood as an approximation, theaccuracy of which we cannot yet quantify since we do not have an answerto the question, “How close to a product state can one prepare a bipartitesystem composed of some type of matter and a collection of gravitons?”This is a question of interest for future work.The influence functional describing a bath of initially thermal gravitonsis then given by the functional integralF [φ] =∫dhTTf∫dhTTi δ(hTTf − hTT ′f ) (3.14)×∫ hTTfhTTiDhTT∫ hTTihTT ′iDhTT ′′ eiSg [hTT ,φ]−SE [hTT ].The integration can be done straightforwardly as again all integrals aregaussian. The resulting influence phase iΦ[φ] for linear gravity isiΦ[φ] = i∫ tftid4x∫ x0tid4x˜[Tµν(x)− T ′µν(x)](3.15)×([Tσρ(x˜) + T′σρ(x˜)]γµνσρ(x− x˜) + i [Tσρ(x˜)− T ′σρ(x˜)] ηµνσρ(x− x˜)).The dissipation and noise kernels areγµνσρ(x) =12M2P∫d3p(2pi)3eip·x Πµνσρ(p)sin px0p, (3.16)ηµνσρ(x) =12M2P∫d3p(2pi)3eip·x coth(βp2)Πµνσρ(p)cos px0p. (3.17)463.2. Thermal Graviton Bath Influence FunctionalThe physical (gauge-invariant) reduced density matrix for matter interactingwith a thermal bath of gravitons is thenρφ[hLf , hTf , φf ] =∫dφiρ˜φ[φi] (3.18)×(δ(Hˆ) UˆG∫ φfφiDφ eiSM [φ]+iSSG[φ]+iΦ[φ] Uˆ †G δ†(Hˆ)).These influence functionals, Eqs. (3.8) and Eq. (3.15) fully describe theinteraction between the environment and the central system, and thus formthe basis for a discussion of decoherence. With the IF one could derive amaster equation, or a quantum Langevin equation, or attempt to approx-imate the functional integral itself. These next steps will be the subjectof future work when we choose a specific matter system and compute itsdecoherence rate.It must be noted that the above influence functional has been previouslyderived in [52] however they incorrectly imposed TT gauge in the path-integral. This is a gauge choice which can only be consistently made invacuum and not in the presence of matter. Moreover, as we will discussfurther in the following section, it is incorrect to simply impose a gauge con-dition on the path integral without following the FP procedure as we haveand integrating over gauge equivalent boundary data. Naively imposing agauge condition leads to a gauge dependent result as evidenced by the dis-agreement between [15, 49, 50, 52]. Mistakes aside, the influence functionalreported in [52] is indeed in agreement with ours (up to a factor of 2). Tounderstand why they obtain the correct influence functional while makingan incorrect gauge-choice we need to return to the evaluation of the pathintegral expression for the propagator (2.63). The modified FP trick weused forced us to integrate over gauge-equivalent boundary data. Given afield hµν the only gauge invariant parts are the TT components7. Since theonly boundary data which is unaffected is that for the TT components, it isthe TT components which end up constituting an environment, and thus itis the TT components which get traced over and it is the TT componentswhich contribute to the influence functional. Integrating over gauge equiva-lent boundary data then implied that we integrate over the boundary datafor hµ0 and hLjk. Without fixed boundary data these variables were actingsimply as Lagrange multipliers in the action which generated the “gauge7The transverse-trace part is also gauge invariant, however it is constrained by theHamiltonian constraint and thus cannot be considered as independent variable473.3. Importance of Gauge Invarianceinvariant dressing” operators and the self-gravity interaction terms. By in-correctly enforcing TT-gauge the authors discarded the terms necessary tomake the matter state gauge invariant (i.e. the dressing operators) andalso discarded the terms which describe the self-gravitational interaction forthe matter, but they retained the essential terms which described the TTradiation i.e the environment.In the limit that the matter is a collection of point particles, Eq. (3.15)has been reported in [49, 50] but in both of these situations it was assumedthat the metric perturbation was purely TT. As a result they missed outon the part of the field constrained by the matter but still obtained thecorrect influence functional. In neither reference are the gauge invarianceand constraints discussed. It is the manifestly gauge invariant computationthat we have done which makes it clear how this computation is to becorrectly done.The constraints of the theory have indeed been properly treated in [51,53], however both computations were done using the canonical linearizedADM formalism rather than using the functional integral approach. Thecomputation done in [51] was in a fixed gauge, while the computation donein [53] was manifestly gauge invariant. They ensured gauge invariance byusing the Dirac formalism. Since the path integral formalism developedhere has been demonstrated as equivalent to the Dirac formalism, we canregard our result as the path integral equivalent of [53]. The strength of ourapproach is that functional integrals make certain approximation schemesmuch more convenient, e.g. semi-classical and eikonal techniques may beemployed.3.3 Importance of Gauge InvarianceThe requirement of gauge invariance led us to correctly identify the correctindependent degrees of freedom and separate them from the variables whichwere constrained. This was essential because in a gauge fixed computationone could mistakenly treat all components of the gauge field as the environ-ment and arrive at a qualitatively different influence functional. A resultrecently reported in the literature suffers from this error [15]. In this sectionwe will intentionally make this error and show that we reproduce the resultof [15].Recall how gauge invariance was maintained in the path integral. Whenwe used the FP trick the integral over the gauge group for intermediate timesfactored out as a constant. However it was an essential point that there483.3. Importance of Gauge Invariancewere remaining integrals over all gauge equivalent boundary data. This isa distinct difference from the way the FP trick is used in the computationof a partition function or generating functional since in both cases thereis no fixed boundary data. In these situations, if the FP determinant alsofactors out the result of the FP trick is to simply insert a gauge fixing deltafunction in the path integral and this delta function is often rewritten as agauge fixing term in the action.In [15] the author made an essential mistake which led to a predictionthat the influence functional depended on every component of the matterstress tensor rather than just the TT components. They handled the gaugeinvariance of the action by merely inserting the harmonic gauge fixing termin the action. As a result they obtained a qualitatively different result fromours. The addition of the harmonic gauge fixing term broke the gauge in-variance of the theory making all components of hµν seemingly independent.If we intentionally make this mistake we can reproduce their results. Thegravitational action Eq. (1.27) can be rewrittenS =∫ tftid4x(−12∂σhµν∂σhµν + ∂µhµν∂σhσν + LM + 1MPhµνTµν),(3.19)where the overline denotes trace reversal hµν = hµν − 12ηµνh. The additionof the harmonic gauge fixing termSgf = −∫ tftid4x∂µhµν∂σhσν (3.20)cancels the second term in Eq. (3.19). In this gauge fixed theory the prop-agator isK(hf , φf ;hi, φi) =∫ φfφiDφ∫ hfhiDhµν eiSM+i∫ tftid4x(− 12∂σhµν∂σhµν+1MPhµνTµν).(3.21)The path integral over hµν can be immediately evaluated,K(hf , φf ;hi, φi) =∫ φfφiDφ eiSM+i(S+Sgf )[hµνi →hµνf ,φ]. (3.22)Unlike the gauge invariant propagator Eq. (2.80) this propagator treats allcomponents of hµν on equal footing. Clearly this propagator maps betweenstates in the full configuration space ψ[hµν , φ] rather than projecting ontothe physical subspace. In the full configuration space all components of the493.3. Importance of Gauge Invariancegauge field are independent variables and a state in which the matter isuncorrelated with the gauge field would be written as a product ρ[h, φ] =ρg[hµν ]ρφ[φ]. Considering the matter as the central system the full gaugefield would then seem to be the environment, and a partial trace would beover all of hµν . An uncorrelated state with a thermal gauge field would thenbe of the formρ[h, φ] = ρφ[φ]∫ hh′Dh′′µν e−(S+Sgf )E [h′′µν ]. (3.23)For such an initial state the reduced density matrix at a later time is givenby the path integralρφ[φf ] =∫dφiρ[φi]∫ φfφiDφ eiSM [φ]+iΦ[φ], (3.24)where the influence phase for the gauge fixed theory is of precisely thesame form as in the gauge invariant theory Eq. (3.15) but with the es-sential difference that in the dissipation and noise kernels Eq. (3.16) and(3.17) the TT projection operator is replaced by the trace-reversal operatorPµνσρ = 12(ηµσηµρ + ηµρηνσ − ηµνησρ). The noise and dissipation kernelscan then be written in terms of the scalar kernels, γµνσρgf = 2PµνσρD andηµνσρgf = 2PµνσρN , whereD(x) =14M2P∫d3p(2pi)3eip·xsin px0p, (3.25)N(x) =14M2P∫d3p(2pi)3eip·x coth(βp2)cos px0p. (3.26)To make contact with reference [15] we can take the path integral repre-sentation of the reduced density matrix evolution and determine the masterequation. The process of passing from a path integral to a master equationin this situation is computationally no different than the tetbook derivationof the Schro¨dinger equation from the path integral in ordinary quantum me-chanics [82]. See Appendix C for the derivation. To lowest order in M−1P ,the master equation satisfied by the reduced density matrix in Eq. (3.24) is∂ρ∂t= −i[HˆM , ρ(t)] (3.27)−∫ ttidx˜0∫d3xd3x˜(ηµνσρ(x− x˜)[Tˆµν(x), [Tˆσρ(x˜), ρ(t)]]−iγµνσρ(x− x˜)[Tˆµν(x), {Tˆµν(x˜), ρ(t)}]).503.3. Importance of Gauge InvarianceIf we choose the kernels from the gauge invariant theory, we reproduce equa-tion (19) of [53] (up to a relative sign between the dissipation and noiseterms). However if instead we choose the kernels from the naively gaugefixed theory the above master equation exactly reproduces equation (17) of[15].Rather than depending only on the TT components of the stress-tensorthe influence functional (and therefore the master equation) in this naivelygauge fixed theory depends on all of the components of the stress tensor.The naively gauge fixed theory has thus made a qualitatively different pre-diction from our manifestly gauge invariant theory. In the limit that thematter is moving non-relativistically the (00) component of the stress ten-sor dominates the other components. This component is essentially themass density of the system, and an influence functional which depends onT 00 would suggest that systems with sufficiently large mass would decohereregardless of the dynamics and shape of the object. This is the conclusiondrawn in [15] which we now claim to be incorrect due to the mistreatmentof the gauge invariance/constraints of the theory. The correct computationof the decoherence rates of a number of example systems will be the topicof future works. For now, a general statement can be made as a correc-tion of this mistaken conclusion. Since the correct gauge invariant influencefunctional depends on the TT part of the stress tensor rather than the (00)components, it should be the change in the mass quadrupole moment notthe mass monopole moment which is most important in quantum gravita-tional decoherence. Such a statement should not be surprising as the timederivative of the mass quadrupole moment is a quantity which of centralimportance in the classical theory of gravitational wave emission.51Chapter 4ConclusionIn this thesis we developed a tool which will be useful to study decoherencein quantum gravity. We used an effective field theory approach which isexpected to capture the universal low every behaviour of whatever the UVcompletion of quantum gravity may be. In particular, we computed theFeynman-Vernon influence functional for a matter system interacting witha bath of thermal gravitons.There have been a number of calculations of the influence functional (orthe related master equation) in linearized quantum gravity in the literature.Of all these references only [51, 53] include an appropriate discussion ofthe fact that quantum gravity is a constrained theory. Furthermore only[53] has a manifestly gauge invariant computation. They use the canoni-cal formalism to compute a master equation for the reduced density matrixfor matter interacting with a thermal bath of gravitons. The discussionof constraints and gauge invariance is of essential importance because dif-ferent gauge-fixed results have been reported in the literature and thereare qualitative disagreements between them depending on the gauge choice.Until now it has not been clear which result to trust and why the differ-ent results don’t agree. Although the master equation presented in [53] isgauge invariant their approach is limited to the canonical quantization for-malism which lacks clear approximation schemes. In this paper we studiedquantum gravitational decoherence using the functional integral formalism,which invites future computations to use functional integral approximationtechniques such as diagrammatic, semi-classical, and eikonal expansions.We presented the first manifestly gauge invariant computation of the re-duced density matrix propagator for matter interacting with a thermal bathof gravitons in a path integral representation. En route we developed a man-ifestly gauge invariant representation of the transition amplitude betweengauge field configurations in the presence of matter. We also demonstratedhow the first-class constraints in both QED and linearized gravity emerge asnatural consequences of gauge invariance in the path integral representationof the propagator. The entire approach was within the functional integralframework, and we demonstrated that our approach is the path integral52Chapter 4. Conclusionequivalent of the Dirac quantization of first-class constrained systems. Ourresult verifies the validity of the influence functional and master equationwhich depend only on the TT parts of the matter’s stress-energy tensor. Af-ter obtaining this main result we explained the mistake made in [15] whichled the author to derive a master equation primarily depending on the massdensity T 00.Our manifestly gauge invariant computation led to the following formalstatements which when actually applied to some specific examples providean illuminating physical picture. The theories we studied are gauge theoriesso the physical states must be invariant under certain gauge transformations.To be invariant under gauge transformations it is sufficient for the states tobe annihilated by the generators of gauge transformation. This conditionprovides us with functional differential equations that the states must satisfywhich can be seen as the constraint equations of the canonical framework.Thus the wavefunctionals for physical states cannot be arbitrary functionalson the configuration space. Some of the gauge field degrees of freedomare independent but there are other (constrained) degrees of freedom forwhich the form of the wavefunctional is determined entirely by the constraintequation, i.e. the requirement of gauge invariance.In the QED example, the transverse part of the field is gauge invari-ant and thus an independent degree of freedom whereas the dependence ofphysical wavefunctionals on the longitudinal part of the field is determinedentirely by the Gauss law constraint. In order for physical states to sat-isfy Gauss’ law, the charged matter is always dressed by a coherent state ofthe electric field corresponding the appropriate Coulomb field. The physicalpicture is not matter interacting with a vector boson field, but rather it ismatter with its accompanying Coulomb dressing that interacts with a fieldconsisting of two types of transverse polarized photons.The gravity example is quite similar. The analogue of Gauss’ law (themomentum constraint) constrains the dependence on the longitudinal partof the field. This constraint similarly requires the matter to be dressed bya coherent gravitational field. In gravity there is also the Hamiltonian con-straint which constrains the configuration space variables, effectively elim-inating the trace-part of the field. The remaining TT degrees of freedomare independent. The physical picture is analogous to QED; we should notthink of matter interacting with a symmetric rank-2 tensor field but ratheras matter (dressed by its appropriate coherent gravitational field) interactingwith two fields corresponding to the two TT polarizations of the graviton.In future works we plan to apply these results to problems of quantumdecoherence by considering specific physical systems and studying the dy-53Chapter 4. Conclusionnamics of their reduced density matrices and computing decoherence rates.Interesting applications include the harmonic oscillator, an extended bodywhich consists of particles coupled via harmonic potentials, a uniformly ac-celerated particle, as well as corrections to the 2 → 2 scattering of matterexcitations. Another further application of the influence functional is to theresurging field of infrared gauge theory physics and its potential relation tothe black hole information paradox. It has recently been demonstrated thatthe emission of soft gauge bosons from a scattering process can lead to anoutgoing state which is almost completely diagonal in the momentum ba-sis [92]. This is inline with the idea that soft bosons may skirt the no-hairtheorem and resolve the black hole information paradox by storing informa-tion holographically [93]. To obtain the influence functional we traced overall outgoing states of the photon/graviton fields, including infinitely longwavelength modes. The influence functional may then be used as a tool forunderstanding information loss due to soft bremsstrahlung in addition tothe information loss expected from environmental fluctuations.54Bibliography[1] N V Prokof’ev and P C E Stamp. Theory of the spin bath. Reports onProgress in Physics, 63(4):669, 2000.[2] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanicaldescription of physical reality be considered complete? Phys. Rev.,47:777–780, May 1935.[3] Mark Van Raamsdonk. Building up spacetime with quantum entangle-ment. International Journal of Modern Physics D, 19(14):2429–2435,2010.[4] D. Hanneke, S. Fogwell Hoogerheide, and G. Gabrielse. Cavity controlof a single-electron quantum cyclotron: Measuring the electron mag-netic moment. Phys. Rev. A, 83:052122, May 2011.[5] S. W. Hawking. Breakdown of predictability in gravitational collapse.Phys. Rev. D, 14:2460–2473, Nov 1976.[6] F. Karolyhazy. Gravitation and quantum mechanics of macroscopicobjects. Il Nuovo Cimento A (1965-1970), 42(2):390–402, Mar 1966.[7] Roger Penrose. On gravity’s role in quantum state reduction. GeneralRelativity and Gravitation, 28(5):581–600, May 1996.[8] T. W. B. Kibble. Relativistic models of nonlinear quantum mechanics.Communications in Mathematical Physics, 64(1):73–82, Dec 1978.[9] T W B Kibble and S Randjbar-Daemi. Non-linear coupling of quantumtheory and classical gravity. Journal of Physics A: Mathematical andGeneral, 13(1):141, 1980.[10] L. Dio´si. Gravitation and quantum-mechanical localization of macro-objects. Physics Letters A, 105(4):199 – 202, 1984.[11] L. Dio´si. A universal master equation for the gravitational violation ofquantum mechanics. Physics Letters A, 120(8):377 – 381, 1987.55Bibliography[12] L. Dio´si. Models for universal reduction of macroscopic quantum fluc-tuations. Phys. Rev. A, 40:1165–1174, Aug 1989.[13] GianCarlo Ghirardi, Renata Grassi, and Alberto Rimini. Continuous-spontaneous-reduction model involving gravity. Phys. Rev. A, 42:1057–1064, Aug 1990.[14] P C E Stamp. Rationale for a correlated worldline theory of quantumgravity. New Journal of Physics, 17(6):065017, 2015.[15] M. P. Blencowe. Effective field theory approach to gravitationally in-duced decoherence. Phys. Rev. Lett., 111:021302, Jul 2013.[16] Richard P Feynman, Fernando B Morinigo, and William G Wag-ner. Feynman lectures on gravitation. European Journal of Physics,24(3):330, 2003.[17] Nathan Rosen. The relation between classical and quantum mechanics.American Journal of Physics, 32(8):597–600, 1964.[18] C. Kiefer. Decoherence in Quantum Field Theory and Quantum Grav-ity, pages 181–225. Springer Berlin Heidelberg, Berlin, Heidelberg,2003.[19] Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P. Singh, andHendrik Ulbricht. Models of wave-function collapse, underlying theo-ries, and experimental tests. Rev. Mod. Phys., 85:471–527, Apr 2013.[20] A. Bassi, A. Großardt, and H. Ulbricht. Gravitational Decoherence.June 2017. arXiv:quant-ph/1706.05677.[21] Bogdan Mielnik. Generalized quantum mechanics. Communications inMathematical Physics, 37(3):221–256, Sep 1974.[22] G. C. Ghirardi, A. Rimini, and T. Weber. Unified dynamics for micro-scopic and macroscopic systems. Phys. Rev. D, 34:470–491, Jul 1986.[23] Donald Marolf. Unitarity and holography in gravitational physics.Phys. Rev. D, 79:044010, Feb 2009.[24] Donald Marolf. Emergent gravity requires kinematic nonlocality. Phys.Rev. Lett., 114:031104, Jan 2015.56Bibliography[25] R. Arnowitt, S. Deser, and C. W. Misner. Dynamical structure anddefinition of energy in general relativity. Phys. Rev., 116:1322–1330,Dec 1959.[26] G. ’t Hooft. Dimensional Reduction in Quantum Gravity. October1993. arXiv:gr-qc/9310026.[27] Leonard Susskind. The world as a hologram. Journal of MathematicalPhysics, 36(11):6377–6396, 1995.[28] Juan Maldacena. The large-n limit of superconformal field theories andsupergravity. International Journal of Theoretical Physics, 38(4):1113–1133, Apr 1999.[29] Edward Witten. Anti de sitter space and holography. Advances inTheoretical and Mathematical Physics, 2(2):253–291, 1998.[30] L. Susskind and E. Witten. The Holographic Bound in Anti-de SitterSpace. May 1998. arXiv:hep-th/9805114.[31] S. W. Hawking. The unpredictability of quantum gravity. Communi-cations in Mathematical Physics, 87(3):395–415, Dec 1982.[32] S. W. Hawking. Wormholes in spacetime. Phys. Rev. D, 37:904–910,Feb 1988.[33] John Ellis, Subhendra Mohanty, and Dimitri V. Nanopoulos. Quan-tum gravity and the collapse of the wavefunction. Physics Letters B,221(2):113 – 119, 1989.[34] S.W. Hawking and R. Laflamme. Baby universes and the non-renormalizability of gravity. Physics Letters B, 209(1):39 – 41, 1988.[35] I. Pikovski, M. Zych, F. Costa, and Cˇ. Brukner. Universal decoherencedue to gravitational time dilation. Nature Physics, 11:668–672, August2015.[36] Yuri Bonder, Elias Okon, and Daniel Sudarsky. Can gravity accountfor the emergence of classicality? Phys. Rev. D, 92:124050, Dec 2015.[37] L. Dio´si. Centre of mass decoherence due to time dilation: paradoxicalframe-dependence. July 2015. arXiv:quant-ph/1507.05828.57Bibliography[38] Y. Bonder, E. Okon, and D. Sudarsky. Questioning universal decoher-ence due to gravitational time dilation. Nature Physics, 12:2, January2016.[39] S. L. Adler and A. Bassi. Gravitational decoherence for mesoscopicsystems. Physics Letters A, 380:390–393, January 2016.[40] Igor Pikovski, Magdalena Zych, Fabio Costa, and aslav Brukner. Timedilation in quantum systems and decoherence. New Journal of Physics,19(2):025011, 2017.[41] Serge Reynaud, Brahim Lamine, Astrid Lambrecht, Paulo Maia Neto,and Marc-Thierry Jaekel. Hyper and gravitational decoherence. Gen-eral Relativity and Gravitation, 36(10):2271–2288, Oct 2004.[42] Brahim Lamine, Re´my Herve´, Astrid Lambrecht, and Serge Reynaud.Ultimate decoherence border for matter-wave interferometry. Phys.Rev. Lett., 96:050405, Feb 2006.[43] W. L. Power and I. C. Percival. Decoherence of quantum wave packetsdue to interaction with conformal space–time fluctuations. Proceedingsof the Royal Society of London A: Mathematical, Physical and Engi-neering Sciences, 456(1996):955–968, 2000.[44] Charles H-T Wang, Robert Bingham, and J Tito Mendona. Quan-tum gravitational decoherence of matter waves. Classical and QuantumGravity, 23(18):L59, 2006.[45] Ertan Go¨klu¨ and Claus La¨mmerzahl. Metric fluctuations and the weakequivalence principle. Classical and Quantum Gravity, 25(10):105012,2008.[46] Heinz-Peter Breuer, Ertan Go¨klu¨, and Claus La¨mmerzahl. Met-ric fluctuations and decoherence. Classical and Quantum Gravity,26(10):105012, 2009.[47] L. H. Ford. Gravitons and light cone fluctuations. Phys. Rev. D,51:1692–1700, Feb 1995.[48] V. A. De Lorenci and L. H. Ford. Decoherence induced by long wave-length gravitons. Phys. Rev. D, 91:044038, Feb 2015.[49] C. Anastopoulos. Quantum theory of nonrelativistic particles interact-ing with gravity. Phys. Rev. D, 54:1600–1605, Jul 1996.58Bibliography[50] Z. Haba and H. Kleinert. Quantum-liouville and langevin equationsfor gravitational radiation damping. International Journal of ModernPhysics A, 17(26):3729–3736, 2002.[51] C Anastopoulos and B L Hu. A master equation for gravitational de-coherence: probing the textures of spacetime. Classical and QuantumGravity, 30(16):165007, 2013.[52] F. Suzuki and F. Queisser. Environmental gravitational decoherenceand a tensor noise model. Journal of Physics: Conference Series,626(1):012039, 2015.[53] T. Oniga and C.H.-T. Wang. Quantum gravitational decoherence oflight and matter. Phys. Rev. D, 93:044027, Feb 2016.[54] Robert M. Wald. General Relativity, chapter 4.4, pages 79, 80. Univer-sity of Chicago Press, 1984.[55] P.A.M. Dirac. Generalized hamiltonian dynamics. Can. J. Math, 2:129–148, 1950.[56] P.A.M. Dirac. Lectures on Quantum Mechanics. Belfer Graduate Schoolof Science, Yeshiva University, NY, 1964.[57] M. Kramer, I. H. Stairs, R. N. Manchester, M. A. McLaughlin,A. G. Lyne, R. D. Ferdman, M. Burgay, D. R. Lorimer, A. Possenti,N. D’Amico, J. M. Sarkissian, G. B. Hobbs, J. E. Reynolds, P. C. C.Freire, and F. Camilo. Tests of general relativity from timing the doublepulsar. Science, 314(5796):97–102, 2006.[58] John Antoniadis, Paulo C. C. Freire, Norbert Wex, Thomas M. Tauris,Ryan S. Lynch, Marten H. van Kerkwijk, Michael Kramer, Cees Bassa,Vik S. Dhillon, Thomas Driebe, Jason W. T. Hessels, Victoria M. Kaspi,Vladislav I. Kondratiev, Norbert Langer, Thomas R. Marsh, Maura A.McLaughlin, Timothy T. Pennucci, Scott M. Ransom, Ingrid H. Stairs,Joeri van Leeuwen, Joris P. W. Verbiest, and David G. Whelan. Amassive pulsar in a compact relativistic binary. Science, 340(6131),2013.[59] Clifford M. Will. The confrontation between general relativity andexperiment. Living Reviews in Relativity, 17(1):4, Jun 2014.59Bibliography[60] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese,K. Ackley, C. Adams, et al. Observation of gravitational waves from abinary black hole merger. Phys. Rev. Lett., 116:061102, Feb 2016.[61] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese,K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al.Tests of General Relativity with GW150914. Physical Review Letters,116(22):221101, June 2016.[62] M. Fierz and W. Pauli. On relativistic wave equations for particlesof arbitrary spin in an electromagnetic field. Proceedings of the RoyalSociety of London A: Mathematical, Physical and Engineering Sciences,173(953):211–232, 1939.[63] R. P. Feynman. Quantum theory of gravitation. Acta Phys. Polon.,24:697–722, 1963.[64] Bryce S. DeWitt. Quantum theory of gravity. ii. the manifestly covari-ant theory. Phys. Rev., 162:1195–1239, Oct 1967.[65] Bryce S. DeWitt. Quantum theory of gravity. iii. applications of thecovariant theory. Phys. Rev., 162:1239–1256, Oct 1967.[66] Gerard ’t Hooft and M. J. G. Veltman. One loop divergencies in thetheory of gravitation. Ann. Inst. H. Poincare Phys. Theor., A20:69–94,1974.[67] Steven Weinberg. Ultraviolet Divergences in Quantum Theories ofGravitation. In General Relativity: An Einstein Centenary Survey,pages 790–831. 1980.[68] John F. Donoghue. Leading quantum correction to the newtonian po-tential. Phys. Rev. Lett., 72:2996–2999, May 1994.[69] John F. Donoghue. General relativity as an effective field theory: Theleading quantum corrections. Phys. Rev. D, 50:3874–3888, Sep 1994.[70] N. E. J. Bjerrum-Bohr, John F. Donoghue, and Barry R. Holstein.Quantum corrections to the schwarzschild and kerr metrics. Phys. Rev.D, 68:084005, Oct 2003.[71] Cliff P. Burgess. Quantum gravity in everyday life: General relativityas an effective field theory. Living Reviews in Relativity, 7(1):5, 2004.60Bibliography[72] John F. Donoghue. The effective field theory treatment of quantumgravity. AIP Conference Proceedings, 1483(1):73–94, 2012.[73] R.P. Feynman and F.L. Vernon Jr. The theory of a general quantumsystem interacting with a linear dissipative system. Annals of Physics,281(12):547 – 607, 2000.[74] Julian Schwinger. Brownian motion of a quantum oscillator. Journalof Mathematical Physics, 2(3):407–432, 1961.[75] L. V. Keldysh. Diagram technique for nonequilibrium processes. Zh.Eksp. Teor. Fiz., 47:1515–1527, 1964. [Sov. Phys. JETP20,1018(1965)].[76] James W. York. Role of conformal three-geometry in the dynamics ofgravitation. Phys. Rev. Lett., 28:1082–1085, Apr 1972.[77] G. W. Gibbons and S. W. Hawking. Action integrals and partitionfunctions in quantum gravity. Phys. Rev. D, 15:2752–2756, May 1977.[78] S.W. Hawking. The path-integral approach to quantum gravity. InS.W. Hawking and W. Israel, editors, General Relativity: An EinsteinCentenary Survey, chapter 15, pages 746–789. Cambridge UniversityPress, Cambridge, 1979.[79] David J. Gross and Roman Jackiw. Low-energy theorem for gravitonscattering. Phys. Rev., 166:1287–1292, Feb 1968.[80] F. Mattei, C. Rovelli, S. Speziale, and M. Testa. From 3-geometrytransition amplitudes to graviton states. Nuclear Physics B, 739(1):234– 253, 2006.[81] G.C. Rossi and M. Testa. The structure of yang-mills theories in thetemporal gauge. Nuclear Physics B, 163:109 – 132, 1980.[82] R. P. Feynman and A. R. Hibbs. Quantum Mechanics and Path Inte-grals. McGraw-Hill, New York, 1965.[83] L.D. Faddeev and V.N. Popov. Feynman diagrams for the yang-millsfield. Physics Letters B, 25(1):29 – 30, 1967.[84] G.C. Rossi and M. Testa. The structure of yang-mills theories in thetemporal gauge. Nuclear Physics B, 176(2):477 – 499, 1980.[85] L. D. Faddeev. The feynman integral for singular lagrangians. Theo-retical and Mathematical Physics, 1(1):1–13, 1969.61[86] Morton Hamermesh. Group Theory and its Application to PhysicalProblems, chapter 8.6, pages 296, 297. Addison-Weslry Publishing Com-pany, Inc., 1962.[87] P. A. M. Dirac. Gauge-invariant formulation of quantum electrodynam-ics. Canadian Journal of Physics, 33(11):650–660, 1955.[88] K. Kuchar. Ground state functional of the linearized gravitational field.J. Math. Phys., 11:3322–3334, 1970.[89] James B. Hartle. Ground-state wave function of linearized gravity.Phys. Rev. D, 29:2730–2737, Jun 1984.[90] A. O. Caldeira and A. J. Leggett. Path integral approach to quantumbrownian motion. Physica A: Statistical Mechanics and its Applications,121(3):587–616, 1983.[91] C. Anastopoulos and A. Zoupas. Nonequilibrium quantum electrody-namics. Phys. Rev. D, 58:105006, Oct 1998.[92] D. Carney, L. Chaurette, D. Neuenfeld, and G. W. Semenoff. Infraredquantum information. June 2017. arXiv:hep-th:1706.03782.[93] A. Strominger. Black Hole Information Revisited. June 2017.arXiv:hep-th/1706.07143.[94] F. Mandl and G. Shaw. Quantum Field Theory, chapter 5.3, pages81–83. Wiley; 2 edition, 2010.62Appendix AThe Classical Action in QEDand Linear GravityIn Chapter 2 we evaluated path integrals over the gauge fields Aµ and hµν .A major point of this thesis was that not all variables are independent andas a result of the gauge invariance of the action we ended up integratingover gauge equivalent boundary data. The only dynamical path integralswhich we had to evaluate were over the transverse components ATj and TTcomponents hTTj . The evaluation of these integrals is identical for the twotheories so here we will just evaluate the QED integral. The relevant integralwe are interested in evaluating is found in Eq. (2.45)∫ ATfATiDAT eiS[AT ,φ]. (A.1)The integrals for the free theory in both real time tf − ti and Euclidean timeiβ are special cases of the above.As mentioned when this integral was first seen, it can be evaluated byshifting the integration variable by a function which extremizes the action.That is, the solution to the classical equation of motion subject to the bound-ary condition that at times ti, tf the classical solution matches the boundarydata of the path integral. The integral over fluctuations is then independentof the boundary data. Since the action is quadratic in the fields the integralover fluctuations simply returns a functional determinant which is an overallconstant irrelevant for our purposes. The result of the integral is then justexp iScl where Scl is the action evaluated for the classical path.The action for the transverse photons is Eq. (2.41)S[AT , φ] =∫ tftid4xPµν(−12∂σAµ∂σAν +AµJν). (A.2)The transverse projector can be written as a sum over orthonormal trans-verse polarization basis vectors, Pµν =∑2λ=1 λµλν . For a 4-vector vµ we63Appendix A. The Classical Action in QED and Linear Gravitydefine the transverse part vλ = λµvµ. The action for transverse photons canthen be written asS[AT , φ] =2∑λ=1∫ tftid4x(−12∂σAλ∂σAλ +AλJ λ). (A.3)The vector components are decoupled and the action is just the sum ofactions for two independent massless scalar fields Aλ coupled to their re-spective sources J λ. Performing a Fourier decomposition on the spatialvariables the action can be writtenS[AT , φ] =2∑λ=1∫ tfti∫d3p(2pi)3(12|∂tAλ|2 − 12p2|Aλ|2 +Aλ(p)J λ(−p)).(A.4)This is precisely the sum of actions for a continuum of decoupled harmonicoscillators Aλ(p) with unit mass, frequencies ωλ(p) = |p|, each coupled toa force J λ(p). The evaluation of the action and thus the path integral fora harmonic oscillator is a textbook exercise but for completeness we willbriefly review it [82].The classical equation of motion for a given mode is(∂2t + |p|2)Aλ = J λ. (A.5)The general solution to this equation is the sum of a homogeneous solutionand an inhomogenous solution obtained by integrating J λ with the retardedGreen’s function. The coefficients of integration are fixed by requiring thesolution to satisfy the boundary conditions Aλ(ti,f ) = Aλi,f . The solutionsatisfying these boundary conditions isAλcl(p) =Aλfsin pTsin p(t− ti) + Aλisin pTsin p(tf − t) (A.6)+∫ ttidsJ λ(s)sin p(t− s)p− sin p(t− ti)sin pT∫ tftidsJ λ(s)cos p(t− s)p,where T = tf − ti. Substituting this solution into the action and summing64Appendix A. The Classical Action in QED and Linear Gravityover modes we obtain the expression for the classical actionScl[ATi → ATf , φ] =2∑λ=1∫d3p(2pi)3p2 sin pT((|Aλi |2 + |Aλf |2) cos pT − 2Aλ∗f Aλi(A.7)+2Aλ∗ip∫ tftidtJ λ sin p(tf − t) +2Aλ∗fp∫ tftidtJ λ sin p(t− ti)− 2p2∫ tftidt∫ ttidsJ λ∗(t)J λ(s) sin p(tf − t) sin p(s− ti)).The generalization to gravity is obvious. The TT projector can be writtenas a sum over orthonormal TT basis tensors Πµνσρ =∑2λ=1 λµνλσρ. Definingthe TT part hλ = λµνhµν , the gravitational action decouples into that of twoscalar fields and the classical action is precisely the same as above if (A,J )are replaced by (h, T ). The action for the free field is the J = 0 case of theabove action and the Euclidean action is obtained from the real time actionby SE = −iS|T=−iβ.65Appendix BInstantaneous InteractionTermsIn the evaluation of the functional integral over the metric variables we foundthat some of the integrals generated a new interaction term for the matterEq. (2.74) which is instantaneous. Such a term arose in a way completelyanalogous to the Coulomb interaction term in QED. In what follows we willshow that this term is indeed the correct interaction potential analogous tothe Coulomb potential.Consider the generating functional for the photon fieldZ[J ] =∫DAµ eiS0[A]+i∫d4xAµJµ , (B.1)where Jµ is a conserved classical current. The functional integral can beperformed and the result computes the vacuum energyE0(∫dt)= i logZ[J ] =12∫d4p(2pi)4JµDµνJν , (B.2)where Dµν(p) = −ηµν/p2 is the time-ordered Green’s function for the elec-tromagnetic field. For compactness we won’t write the pole prescriptionexplicitly but this does not affect the results. Now we will follow a proce-dure done in [94].We want to introduce a convenient orthonormal basis of vectors whereone is purely timelike nµ = (1, 0, 0, 0), one is longitudinal µp = (0,p/|p|), andthe other two are transverse. We will call the transverse vectors µr (r = 1, 2).An explicit expression for µp isµp =pµ + (p · n)nµ(p2 + (p · n)2) 12. (B.3)It can be checked that pµµp = |p| and pµµr = 0. The Minkowski metric canthen be written in terms of this orthonormal basis ηµν = −nµnν + µp νp +∑r µr µr . Since we are interested in the contraction of ηµν with the conserved66Appendix B. Instantaneous Interaction Termscurrent pµJµ = 0 we can expand out these basis vectors and drop any termslinear in pµ. Doing so, the Green’s function can be written asDµν = −ηµνp2= −∑r µr µrp2+nµnνp2 + (p · n)2 . (B.4)Now we recognize the sum over transverse polarization basis vectors as thetransverse projector, and note that p2 + (p · n)2 = |p|2. With the Green’sfunction rewritten in this way the log of the generating functional can bewritten in the nice formE0(∫dt)= −12∫d4pJµPµνJνp2+12∫dt∫d3p(2pi)3J0J0|p| (B.5)= −12∫d4pJµPµνJνp2+12∫dt∫d3xd3yJ0(x)J0(y)4pi|x− y| .What we’ve done is separate the contribution from transverse photons fromthe interaction energy of the source. The first term describes transversephotons and the second term can be identified as −SC , the Coulomb inter-action potential. For a static source we simply find that the energy of theconfiguration is given by the Coulomb potential energy.We can repeat the same exercise for linearized gravity. The generatingfunctional for linearized metric perturbation is given byi logZ[T ] =12∫d4p(2pi)4TµνDµνσρTσρ, (B.6)where Tµν is a conserved classical current and Dµνσρ = −Pµναβ/p2 is thetime-ordered Green’s function for the metric perturbation. The index struc-ture Pµνσρ =12(ηµσηµρ + ηµρηνσ − ηµνησρ) corresponds to the trace-reversaloperator. We can use the above decomposition again for each factor of theMinkowski metric that appears in the trace-reversal operator, and again dis-card all terms linear in pµ because pµTµν = 0. Of course this still leavesmany terms. When contracted with the stress tensor many of these termscan are seen to be duplicates of each other. Finally, the generating functionalmay be written down in this orthonormal basis,i logZ[T ] =− 12M2P∫d4p(2pi)4TµνΠµνσρTσρp2(B.7)+14M2P∫dt∫d3x1∇2(T 00T 00 − 4T 0jPjkT 0k+2T 00PjkTjk +∂0T00∂0T00∇2).67Appendix B. Instantaneous Interaction TermsHaving separated off the contribution from TT gravitons we found a staticinteraction potential. This term can readily be identified as−SSG, Eq. (2.74).We then see that indeed the instantaneous gravitational interaction term isprecisely analogous to the Coulomb interaction in QED.68Appendix CDerivation of the MasterEquation from the PathIntegralIn this section we will review how one derives the Schro¨dinger equationfrom a path integral representation of the propagator. We will then proceedto make only a slight generalization to show how a master equation fora reduced density matrix can be derived from a reduced density matrixpropagator.To derive the Schro¨dinger equation we will need to quickly establish sometextbook results. Firstly, the derivative of the propagator with respect tothe final coordinate is∂xfK =∫ xfxiDx(i∂S∂xf)eiS . (C.1)Since we are taking the quantum average of ∂xfS Ehrenfest’s theorem allowsus to use the equation from classical physics ∂xfS = pf , where p =∂L∂x˙ isthe canonical momentum. Thus, we can write−i∂xfK =∫ xfxiDx pf eiS . (C.2)Secondly, the time derivative of the propagator is∂tfK =∫ xfxiDx(i∂S∂tf)eiS , (C.3)which can be rewritten using the classical equation ∂tfS = −H(xf , pf ),where H = x˙p− L is the Hamiltonian of the system∂tfK = −i∫ xfxiDxH(xf , pf ) eiS . (C.4)69Appendix C. Derivation of the Master Equation from the Path IntegralUsing the momentum relation to re-express pf as a derivative we obtain thedifferential equation satisfied by the propagator∂tfK = −iH(xf ,−i∂xf )K, (C.5)which of course is precisely the Schro¨dinger equation. If the propagatorsatisfies this equation then upon integrating with initial data we find that awavefunction at time tf satisfies the same equation∂ψ(x, t)∂t= −iH(x,−i∂x)ψ(x, t). (C.6)Rather than writing the state in position basis we could write the Schro¨dingerequation as basis independent equation by introducing the abstract Hamilto-nian operator. This gives the general expression of the Schro¨dinger equation∂t|ψ(t)〉 = −iHˆ(xˆ, pˆ)|ψ(t)〉. (C.7)Now suppose we had two distinct particles with coordinate labels x1 andx2, each having its own free action with corresponding free HamiltonianHj(xj , pj), and let the particles be coupled via a retarded interaction. Thepropagator for such a system isK(x1f , x2f , tf ;x1i , x2i , ti) = (C.8)=∫ x1fx1iDx1∫ x2fx2iDx2 eiS1[x1]+iS2[x2]+i∫ tftidt∫ ttidsΦ(x1,x˙1,x2,x˙2;t,s).The above derivation implies that the Schro¨dinger equation for such a systemis then∂ψ(x1, x2, t)∂t=− i (H1(x1, pˆ1) +H2(x2, pˆ2))ψ(x1, x2, t) (C.9)−(∫ ttidsΦ(x1, ˆ˙x1, x2, ˆ˙x2; t, s))ψ(x1, x2, t),where the velocity operators are to be interpreted as functions of xj and pˆj . IfΦ is a function only of coordinates and not velocities then then the situationis simple and the canonical momenta are just that of the free Hamiltonianand the velocities can be solved for as usual. There is a difficulty howeverif Φ is a function of velocities. In that case pj depends on the derivativesof Φ, which in turn depends on pj through the velocities. The expressionof the velocities in terms of the momenta is then an iterative Born series.70Appendix C. Derivation of the Master Equation from the Path IntegralIf the interaction is assumed small we can make a Born approximation andtruncate the iterative equation at lowest order. In this approximation thecanonical momenta are simply given by that of the free Hamiltonian. We’llcall the Born approximated operator Φ0(x1, pˆ1, x2, pˆ2; t, s).To bring us closer to an expression looking like a master equation letsassume that the interaction has the formΦ0(x1, pˆ1, x2, pˆ2; t, s) =(fˆ1(t)− fˆ2(t))D(t, s)(fˆ1(s) + fˆ2(s))(C.10)+i(fˆ1(t)− fˆ2(t))N(t, s)(fˆ1(s)− fˆ2(s))where D(t, s) and N(t, s) are just functions of time and fˆj(t) = f(xˆj , pˆj , t).If we also write Hˆj = H(xˆj , pˆj) then we can expand out the interaction andwrite the Schro¨dinger equation as∂ψ(x1, x2, t)∂t=− i(Hˆ1 + Hˆ2 −∫ ttidsD(t, s)(fˆ1(t)− fˆ2(t))(fˆ1(s) + fˆ2(s))(C.11)− i∫ ttidsN(t, s)(fˆ1(t)− fˆ2(t))(fˆ1(s)− fˆ2(s)))ψ(x1, x2, t)Rather than working in position space this expression can be made morecompact if we invent some notation and write a basis independent expres-sion. Let’s define the operators ρ and fˆ(t) such that the position spacematrix elements are fˆ1(t)ψ(x1, x2, t) = 〈x1|fˆ(t)ρ|x2〉 and fˆ2(t)ψ(x1, x2, t) =〈x1|ρfˆ(t)|x2〉. Depending on which side fˆ acts we can describe either parti-cle. With this step, and the final assumption that H2 = −H1 we can writethe Schro¨dinger equation for this strange system in the basis independentform∂ρ∂t= i[Hˆ(t), ρ]−∫ ttids(N(t, s)[fˆ(t), [fˆ(s), ρ(t)]]−iD(t, s)[fˆ(t), {fˆ(s), ρ(t)}])(C.12)Now, it would take a rather strange system of two particles to be describedby a Hamiltonian which has i) an interaction of this form as well as ii)H2 = −H1. Luckily with this computation we were never really interestedin a system of two real particles. Instead we were interested in the equationof motion satisfied by the reduced density matrix. In this case the twocoordinates x1, x2 do not label different particles, they label the ket and brapart of the density matrix respectively. For a full density matrix there areno terms which “couple” the two paths taken by these coordinates (i.e. the71Appendix C. Derivation of the Master Equation from the Path Integralforward and backward in time paths). We’ve seen though that a genericform of the influence functional is F [φ] = eiΦ[φ], and it certainly couples thepaths. The strange retarded interaction (C.10) is then nothing other thanthe influence phase. We’ve thus seen that a reduced density matrix whichevolves according to a propagator of the form Eq. (C.8), will satisfy themaster equation Eq. (C.12). The generalization from a single coordinate xto a generic set of coordinates φ is immediate.72

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0354259/manifest

Comment

Related Items