{"http:\/\/dx.doi.org\/10.14288\/1.0435666":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Physics and Astronomy, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Waterfield, Conor","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2023-08-30T21:14:46Z","type":"literal","lang":"en"},{"value":"2023","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Stochastic gravity is an extension of semiclassical gravity in which general relativity and quantum field theory are combined using the Schwinger-Keldysh formalism. This approach opens up the ability to study problems where quantum fluctuations are important in semiclassical gravity. This framework has been used to demonstrate the possibility of parametric resonance between fluctuating quantum fields and spacetime causing an accelerated expansion of the universe. This may be one possible solution to the cosmological constant problem.\r\n\r\nIn this thesis, I focus on both understanding the consequences of stochastic gravity models and developing consistent calculation methods. Fluctuating spacetime may indicate the possibility of scattering. Regularization techniques must be used in quantum field theory calculations. A method that involves a UV cutoff and maintains Lorentz invariance may be one useful approach.\r\n\r\nScattering is studied through various approaches; as a form of Rayleigh scattering, as a result of interacting field theories, and as a direct influence on geodesics. Quantum field theory approaches to the scattering of geodesics are determined to be a small effect and are not predicted in the framework of semiclassical gravity or stochastic gravity. The study of geodesic deviation in interferometer experiments gives some indication that parametric resonance effects could have a significant role in stochastic gravity.\r\n\r\nThe possibility of using Pauli-Villars as a Lorentz invariant approach to regularize quantum fields in stochastic gravity calculations is explored. The Pauli-Villars regularization scheme is shown to have significant problems when applied to stochastic gravity, such as failures in second-order stress-energy tensor terms and restrictions due to the kinematic properties of negative-norm fields.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/85706?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"Geodesic Scattering and LorentzInvariance in Stochastic GravitybyConor WaterfieldB.Sc., Saint Mary\u2019s University, 2021A 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 2023\u00a9 Conor Waterfield 2023ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:  Geodesic Scattering and Lorentz Invariance in Stochastic Gravity  submitted by Conor Waterfield  in partial fulfilment of the requirements for the degree of Master of Science in Physics  Examining Committee: William G. Unruh, Professor, Physics and Astronomy, UBC Supervisor  Gordon W. Semenoff, Professor, Physics and Astronomy, UBC Supervisory Committee Member   AbstractStochastic gravity is an extension of semiclassical gravity in which gen-eral relativity and quantum field theory are combined using the Schwinger-Keldysh formalism. This approach opens up the ability to study problemswhere quantum fluctuations are important in semiclassical gravity. Thisframework has been used to demonstrate the possibility of parametric res-onance between fluctuating quantum fields and spacetime causing an accel-erated expansion of the universe. This may be one possible solution to thecosmological constant problem.In this thesis, I focus on both understanding the consequences of stochas-tic gravity models and developing consistent calculation methods. Fluctu-ating spacetime may indicate the possibility of scattering. Regularizationtechniques must be used in quantum field theory calculations. A methodthat involves a UV cutoff and maintains Lorentz invariance may be oneuseful approach.Scattering is studied through various approaches; as a form of Rayleighscattering, as a result of interacting field theories, and as a direct influenceon geodesics. Quantum field theory approaches to the scattering of geodesicsare determined to be a small effect and are not predicted in the framework ofsemiclassical gravity or stochastic gravity. The study of geodesic deviation ininterferometer experiments gives some indication that parametric resonanceeffects could have a significant role in stochastic gravity.The possibility of using Pauli-Villars as a Lorentz invariant approachto regularize quantum fields in stochastic gravity calculations is explored.The Pauli-Villars regularization scheme is shown to have significant prob-lems when applied to stochastic gravity, such as failures in second-orderstress-energy tensor terms and restrictions due to the kinematic propertiesof negative-norm fields.iiiLay SummaryObservations show that the universe is expanding, and that expansion isaccelerating. The mechanism for this is known as \u2018dark energy\u2019, thoughits exact nature is not clear. One possible explanation is that fluctuationscaused by interactions between gravity and the quantum fields that makeup the contents of the universe could allow a very slight expansion thataccelerates over time.In this thesis, I look at understanding the mathematical framework un-der which we can understand these fluctuations called \u2018Stochastic Gravity\u2019.I look at the possibility that these fluctuations can cause the scattering ofwaves traveling across long distances in the universe. I also study an ap-proach to calculating the quantities in stochastic gravity that could be usedin a model of accelerated expansion while still describing the type of universewe think we live in.ivPrefaceThis thesis contains original, unpublished work by the author, ConorWaterfield, under the guidance of Prof. William G. Unruh.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 In This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Semiclassical Gravity . . . . . . . . . . . . . . . . . . . . . . . 52.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Linearized Gravity . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Synchronous Gauge . . . . . . . . . . . . . . . . . . . 72.2.2 Harmonic\/de Donder Gauge . . . . . . . . . . . . . . 72.2.3 Transverse-Traceless Gauge . . . . . . . . . . . . . . . 82.2.4 Issues with Linearized Gravity . . . . . . . . . . . . . 82.3 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . 92.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Quantum Field Theory in Curved Spacetime . . . . . . . . . 142.6 Semiclassical Gravity . . . . . . . . . . . . . . . . . . . . . . 142.6.1 Perturbative Semiclassical Gravity . . . . . . . . . . . 162.6.2 Next-order Contribution . . . . . . . . . . . . . . . . 182.7 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7.1 The Cosmological Constant Problem . . . . . . . . . 222.7.2 Bare Cosmological Constant and Fine-tuning . . . . . 222.7.3 Quintessence . . . . . . . . . . . . . . . . . . . . . . . 242.7.4 Vacuum Fluctuations . . . . . . . . . . . . . . . . . . 252.8 Beyond Semiclassical Gravity . . . . . . . . . . . . . . . . . . 25viTable of Contents3 Stochastic Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Issues with Quantum Gravity . . . . . . . . . . . . . . . . . . 263.2 Schwinger-Keldysh Formalism . . . . . . . . . . . . . . . . . 273.3 Stochastic Gravity from Schwinger-Keldysh Functional For-malism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 The Noise Kernel as a Stochastic Source . . . . . . . . . . . 333.5 Predictions of Stochastic Gravity . . . . . . . . . . . . . . . . 363.5.1 Dark Energy . . . . . . . . . . . . . . . . . . . . . . . 373.5.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . 384 Geodesic Scattering . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Justification for Scattering . . . . . . . . . . . . . . . . . . . 404.1.1 Multiple Scattering . . . . . . . . . . . . . . . . . . . 404.1.2 Rayleigh Scattering . . . . . . . . . . . . . . . . . . . 414.2 Single Scattering with Gravitons . . . . . . . . . . . . . . . . 424.2.1 The Stochastic Gravity Approach . . . . . . . . . . . 434.2.2 Interpretation of Wave Scattering . . . . . . . . . . . 474.3 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . 484.3.1 Experimental Measurement of Geodesic Force . . . . 484.3.2 Intrinsic and Induced Geodesic Deviation . . . . . . . 514.4 Instability in Geodesic Scattering . . . . . . . . . . . . . . . 524.4.1 Parametric Resonance from Stochastic Gravity . . . . 554.5 Conclusion on Geodesic Scattering . . . . . . . . . . . . . . . 575 Pauli-Villars Regularization . . . . . . . . . . . . . . . . . . . 605.1 Issues with Breaking Lorentz Invariance . . . . . . . . . . . . 605.2 Introduction to Pauli-Villars . . . . . . . . . . . . . . . . . . 625.2.1 Regularizing \u03d54 Theory in 2+1D . . . . . . . . . . . . 635.3 Regularizing the Semiclassical Stress-Energy Tensor . . . . . 655.4 Bare Ghost Fields in Curved Spacetime . . . . . . . . . . . . 715.4.1 Negative-norm in the Free-Theory . . . . . . . . . . . 715.4.2 Interaction with the Gravitational Field . . . . . . . . 725.4.3 Scattering Amplitudes of Different Coefficients . . . . 765.4.4 Scalar Field Scattering . . . . . . . . . . . . . . . . . 785.5 The Stress-Energy Tensor in Stochastic Gravity . . . . . . . 795.5.1 Modifying the Lagrangian . . . . . . . . . . . . . . . 815.6 Conclusion on Pauli-Villars . . . . . . . . . . . . . . . . . . . 83viiTable of Contents6 Conclusion and Future Work . . . . . . . . . . . . . . . . . . 856.1 The Lorentz-Invariant Approach . . . . . . . . . . . . . . . . 856.1.1 Using the Einstein-Langevin Equation . . . . . . . . . 866.2 Approaches Without a Cutoff . . . . . . . . . . . . . . . . . 866.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91AppendicesA Metric Perturbation Calculations . . . . . . . . . . . . . . . . 98B Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 104viiiList of Figures4.1 An example diagram of many small single-scattering eventsleading to a wide normal distribution in scattered angle . . . 414.2 Feynman diagrams corresponding to \u27e8\u2126| T {\u03d5(x)\u03d5(y)} |\u2126\u27e9 ex-panded up to O(\u03b24). . . . . . . . . . . . . . . . . . . . . . . . 445.1 Feynman diagrams of various ghost field interactions withgravitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Feynman diagrams contributing to the stress-energy tensorexpectation value for positive and negative fields. . . . . . . . 815.3 Feynman diagrams contributing to the noise kernel for posi-tive and negative fields. . . . . . . . . . . . . . . . . . . . . . 81ixAcknowledgmentsMajor thanks to my supervisor, Prof. William G. Unruh, who gave meopportunities to work on interesting ideas, improve myself as a researcher,and who gave me wise insights into aspects of this research that come withyears of experience. I am grateful to NSERC for funding my research.Thanks to my fellow UBC students Pompey, Rana, Abishek, and Peilinfor their collaboration in my learning of these topics in theoretical physicswhich have formed the foundation for this work. Many thanks to Prof.Gordon W. Semenoff for his helpful insights into approaches towards theresearch in this thesis. I am also thankful to Dr. Qingdi Wang for usefuldiscussion and insight into some of the topics of this work.I would like to acknowledge my parents, David and Angela, and my sisterRo\u00b4is\u00b4\u0131n, for their support which has allowed me to focus on keeping up withmy work. Lastly, I would like to thank my partner Sneha for her continuoussupport in keeping me sane throughout my time at UBC.xDedicationThis thesis is dedicated to my family, David, Angela and Ro\u00b4is\u00b4\u0131n; and toSnehaxiChapter 1IntroductionGeneral Relativity and Quantum Mechanics are the two pillars of modernphysics, though they have fundamental inconsistencies. A theory of quan-tum gravity is the ultimate goal of this area of study, and there are manydifferent approaches [1]. Quantum Gravity exists at energy scales beyondour current particle physics capabilities in the 1013eV range at the LargeHadron Collider [2]. While experimental capabilities are still far from therelevant scales, the highest energy particle ever detected was in the 1020eVrange [3] which is only 6 orders of magnitude lower than the Planck scale.Furthermore, cosmological measurements indicate the presence of dark en-ergy [4], as described by the \u03bbCDM model of cosmology. The theoretical ex-planations for this phenomenon attempt to relate this quantity to a vacuumenergy, though the exact explanation has remained unclear [5]. Ultimately,a quantum theory of gravity may be necessary for a complete explanation.The study of quantum gravity includes many approaches, though themost prominent are \u2019top-down\u2019 [6] including string theory [7] and loop quan-tum gravity [8]. The signature of these theories is that they begin with newphysics at high energy and from them emerges a quantum description ofgravity. The high energy scales of the physics described by these theoriesmake them difficult to verify experimentally, though various astrophysicalclaims may be testable [9, 10].As an alternative approach, much of the progress in understanding theoverlap between quantum mechanics and general relativity in the last half-century has come from \u2019bottom-up\u2019 approaches. The earliest of these camein the form of \u2019Quantum Field Theory in Curved Spacetime\u2019 (QFTCST)[6] which involves the study of quantum fields on curved spacetime back-grounds. This framework produced many of the early results in this fieldincluding the predictions of particle creation as a result of black holes [11],accelerating reference frames [12], and expanding spacetimes [13]. These ap-proaches often made use of non-perturbative approaches such as Bogolubovtransformations to describe the properties of the fields in predeterminedspacetimes. Problems of backreaction and renormalization were largely re-solved, and described by Wald\u2019s axioms [14, 15].1Chapter 1. IntroductionIn the 1980s, semiclassical gravity emerged [6] as an approach that al-lowed for the inclusion of interactions between the quantum fields and themetric. The expectation value of the stress-energy tensor is coupled to thegravitational field using the Einstein Field Equations to maintain the clas-sical nature of gravity [6, 16]. As an approximation, the fluctuations areassumed to be negligible, meaning that the two-point correlation functionof the quantum stress-energy tensor operator is dominated by the squareof the expectation value. In this framework, new cosmological problemscould be approached through the use of systems where interaction betweenthe quantum fields and the gravitational field were both essential. This in-cluded cosmological backreaction problems, most notably the introductionof Cosmic Inflation [17\u201319] which is based on the idea that the high en-ergy density of a field in a vacuum decay could produce a rapid inflationin the early universe. This theory could explain the horizon problem dueto large-scale correlations in the Cosmic Microwave Background (CMB),the apparent flatness of the universe, and also the capability of semiclassi-cal gravity to give insight into cosmological phenomena out of the reach ofexperiment.Despite this success, semiclassical gravity runs into some significantphilosophical and practical issues. Some of the issues of superposition havebeen debated, with some arguments that the interaction between classicaland quantum fields is not possible [20], while others argue that this claimis either out of the reach of experiment [21] or the assumptions about semi-classical gravity are incorrect [22]. On the practical side, fluctuations canbecome significant in many systems, even far from the Planck scale [16].Attempting to calculate vacuum energy densities in semiclassical grav-ity is extremely problematic as the results in the form of the cosmologicalconstant problem [5]. The prediction of semiclassical gravity is 10120 ordersof magnitude higher than the apparent cosmological constant as first deter-mined by Riess et al. [4], and would require significant fine-tuning to makeit sensible. This result is also dependent on the regularization techniqueand cutoff used [23] which deviates from the ideas of axiomatic regulariza-tion as proposed by Wald [14]. Following this principle, the vacuum energyexpectation value would be zero. Alternative explanations include either afine-tuned cosmological constant [24] or other quantum effects that cause anapparent cosmological constant [25, 26].Stochastic gravity represents the next step of the evolution of bottom-upquantum gravity [6]. This formulation, as proposed by B. L. Hu and others[27\u201329] includes the next order of contributions coming from the fluctuationsneglected in the semiclassical approach [16]. Much of this has been expanded21.1. In This Thesison by Hu and Verdaguer [6]. The derivation of this approach comes fromusing the Schwinger-Keldysh formalism [30, 31]. The Schwinger-Keldyshformalism introduces double degrees of freedom with positive and negativetime directions respectively, where the results are taken in the limit thatthey meet at some boundary. In this case, a path integral formulation isused under Feynman-Vernon influence functional formalism [32]. This isalso known as the \u2019In-In\u2019 formalism or Closed Time Path (CTP) formalism.The most useful result of stochastic gravity is the Einstein-Langevinequation which can be used to relate curvature to a stochastic componentof the stress-energy tensor. Both the functional formalism and the Einstein-Langevin equation have been used to study a variety of problems.The study of cosmological backreaction has included the CTP formalism[6,33, 34]. The study of backreaction and weak inhomogeneities is useful forgaining a better understanding of the formation of matter density pertur-bations in the early universe that lead to structure formation. There havebeen uses for the stochastic gravity framework in studying black holes, suchas quantum fluctuations in Hawking flux [35, 36]. It has also been used instudying the structure of relativistic stars where quantum fluctuations canbe important in the stability of their structure [37].Using the stochastic gravity framework, Wang et al. [25] demonstrated amechanism in stochastic gravity that may explain the accelerated expansionof the universe as described by the \u03bbCDM model [4]. By assuming aninhomogeneous FRW-style metric and solving the Einstein field equations,they showed that the solutions for the scale factor obeyed a parametricoscillator equation. By showing that there is a parametric resonance inthe scale factor due to fluctuating quantum fields, it can be shown that thetuning of certain parameters with reasonable values gives an estimate for theaccelerated rate of expansion consistent with observation. Some of this workhas been analyzed further [38, 39] to precisely define the modified axioms ofthis theory compared with the Semiclassical predictions.1.1 In This ThesisIn this thesis, I focus on studying some of the questions that arise in theframework of stochastic gravity, particularly the problem of geodesic scat-tering and the validity of Pauli-Villars-based Lorentz-invariant models.In the second chapter, I discuss the framework of semiclassical gravityto define the mathematical framework and formalisms I will use. The basisof semiclassical gravity is general relativity (GR), which is at the core of all31.1. In This Thesisof the work in this thesis. From GR, one can define linearized gravity, whichallows perturbative approximations at low energies. This will be useful forworking with Quantum Field Theory (QFT), which is often understood per-turbatively. There is some gauge freedom in linearized gravity, and so somerelevant gauges are discussed. The functional formalism and its relation-ship to the formal approach to semiclassical gravity are discussed, and somemethods for regularization are examined. The cosmological constant prob-lem is introduced, and the inconsistencies with the semiclassical approachare discussed.In the third chapter, I discuss the framework of stochastic gravity. Idiscuss some of the motivations, particularly as a \u2019bottom-up\u2019 approach toquantum gravity. I demonstrate the derivation from the Schwinger-Keldyshformalism from a functional approach, demonstrating the extension of semi-classical gravity. I discuss some of the results from this framework with afocus on the prediction of accelerated cosmological expansion due to para-metric resonance.In the fourth chapter, I study the problem of scattering in a stochasticallyfluctuating spacetime. I discuss some of the applications of scattering andits relationship to fluctuating spacetime, particularly Rayleigh scattering. Ithen look at the issues in the standard QFT approach to scattering in tryingto describe scattering due to metric perturbations in both semiclassical andthe standard stochastic gravity approach. I look at some approaches tounderstanding geodesic deviation that has been used to study the quantumeffects of gravitons and the potential relationship to work on the cosmologicalconstant problem.In the fifth chapter, I look at defining Pauli-Villars regularization instochastic gravity. I look at definitions in flat spacetime that can be gener-alized to curved spacetime applications. I show the approach to regularizingsemiclassical gravity with Pauli-Villars to lowest order. I look at the issuesof the kinematics of bare ghost fields. Finally, I discuss the problems withregularizing higher-order contributions used in stochastic gravity and somepotential solutions.In the last chapter, I discuss ways to combine some of the issues andsolutions that have arisen in the previous chapters. I outline some alternativeapproaches to Lorentz-invariant models that may allow calculations startingfrom stochastic gravity formalism. Finally, I present some of the aspects ofthe preceding work that may indicate areas with the potential for furtherinvestigation.4Chapter 2Semiclassical GravityNote: This chapter contains an overview of semiclassical gravity, includ-ing introductions to general relativity, quantum field theory, and cosmology.More detailed overviews for general relativity and cosmology can be found intextbooks such as [40, 41], and for quantum field theory in textbooks such as[42]; in particular, Chapter 9 of [42] for functional methods.2.1 General RelativityThe basis of the modern understanding of gravity, General Relativity (GR),is based on several classical concepts. It is built on the concepts such ascovariance, the equivalence principle, and background independence. Ingeneral relativity, spacetime has a curvature that couples with the stress-energy tensor describing matter and energy. General relativity is based onthe principles of relativity, making it a Lorentz invariant theory. Generalrelativity describes a generally covariant theory in which the physics is in-dependent of the reference frame or coordinates. The equivalence principlestates that all matter couples equally to the gravitational field.GR is formulated independent of background, meaning the metric fullydescribes spacetime on its own. Any perturbations must be defined in termsof a background metric, though this metric has no inherent physical meaningin GR. The most useful background metric one may choose is flat Minkowskispacetime, which is the completely flat vacuum solution corresponding tospecial relativity. Perturbative Quantum Field Theory is generally definedin terms of the Minkowski metric, so it makes a useful background. Thereare other standard metrics that are also useful; the other used frequentlyin this thesis is the Friedmann-Robertson-Walker (FRW) which is used as astandard metric in cosmology.Other spacetimes can be described as having a different \u2019geometry\u2019 givenby a different metric tensor g\u00b5\u03bd . Different geometries are results of the energyand matter content of the universe which are described by the stress-energytensor T\u00b5\u03bd . The geometry can be related to the stress-energy tensor throughthe Einstein Field Equations:52.2. Linearized GravityR\u00b5\u03bd \u2212 12Rg\u00b5\u03bd + \u03bbg\u00b5\u03bd =8\u03c0Gc4T\u00b5\u03bd (2.1)This equation includes the Ricci tensor R\u00b5\u03bd and Ricci scalar R, a non-zero cosmological constant \u03bb and a constant 8\u03c0Gc4which I will denote as \u03bathrough the rest of this thesis. As a convention, I will also use c = \u210f = 1,which means results will be in natural units.In this thesis, I will focus on GR as a field theory defined by an action.The Einstein-Hilbert action is given by:SEH =\u222bd4x\u221a\u2212g(R\u2212 2\u03bb)2\u03ba(2.2)This gives the Einstein Field Equations when the variation is taken withrespect to the metric. Adding in an action corresponding to matter will givethe stress-energy tensor term when it is varied with respect to the metric inthe same way as the gravitational action.General relativity has been a very successful theory used in many areas ofastrophysics and as a starting-off point for many more theories in theoreticalphysics. More recent examples of confirmations of phenomena predicted byGR include the observations of blackholes - the most compelling of which aresupermassive blackholes located in the centers of galaxies for which there isgood evidence for in our own galaxy and with the Event Horizon Telescope[43, 44]. Another such prediction is that of gravitational waves, for whichthere is compelling evidence through the LIGO observatory [45].2.2 Linearized GravityOne method used frequently in GR is perturbation theory, called linearizedgravity. This is useful in cosmology and the study of gravitational waves.Starting from defining the metric as being composed of a perturbationaround a background metric:g\u00b5\u03bd = b\u00b5\u03bd + h\u00b5\u03bd (2.3)Typically, one takes b\u00b5\u03bd = \u03b7\u00b5\u03bd so that there are perturbations on a flatbackground. Part of the usefulness of this approach is that there is at thelowest order a linear relationship with the perturbations, which is valid forsmall perturbations and low energy.Even though there are 16 indices of h\u00b5\u03bd , there are only 6 degrees of free-dom. The metric tensor is symmetric, and so in flat spacetime, this also62.2. Linearized Gravityapplies to the perturbation; h\u00b5\u03bd = h\u03bd\u00b5. One must also consider diffeomor-phisms, which are transformations in the coordinates. In linearized theory,this is called a gauge transformation. Under such a transformation, themetric transforms as:h\u00b5\u03bd \u2192 h\u00b5\u03bd + \u2202\u00b5\u03be\u03bd + \u2202\u03bd\u03be\u00b5 (2.4)One has the freedom to choose a small \u03be such that there is invarianceunder this gauge transformation. As this linearized theory requires smallh\u00b5\u03bd , \u03be must also be small as to maintain linearity. This means four degrees offreedom can be removed, with 6 physical degrees of freedom remaining [40].In the previous equation, one can identify a part h\u00b5\u03bd given in a certain gauge,and a gauge part \u2202\u00b5\u03be\u03bd + \u2202\u03bd\u03be\u00b5 which transforms to a physically-equivalentform h\u00b5\u03bd in a different gauge. As there are four components to \u03be\u00b5, there arefour degrees of freedom that can be fixed to a particular gauge.There are some useful gauges that I will briefly outline.2.2.1 Synchronous GaugeIn this gauge, h0\u00b5 = 0 so that the metric is reduced to the six spatialdegrees of freedom. The metric will then take a form with only free spatialcomponents as a broad generalization of the FRW metric [25]:g\u00b5\u03bd =\uf8eb\uf8ec\uf8ec\uf8ed\u22121 0 0 00 a2 d e0 d b2 f0 e f c2\uf8f6\uf8f7\uf8f7\uf8f8 (2.5)Taking the condition h0\u00b5 + \u22020\u03be\u00b5 + \u2202\u00b5\u03be0 = 0 gives four equations whichgive the four components to \u03be\u00b5. This is equivalent to Gaussian normalcoordinates which describe the coordinates on hypersurfaces \u03a3 which arespacelike, and so the coordinates are defined as perpendicular to the time-like tangent vector n\u00b5. I will not use this gauge in this work, though oneapproach could involve this gauge.2.2.2 Harmonic\/de Donder GaugeThis gauge is defined by a harmonic coordinate condition, which sets\u2202\u00b5(\u221a\u2212gg\u00b5\u03bd) = 0. To the linear order in metric perturbations, this gives thecondition 12\u2202\u03bdh = \u2202\u00b5h\u00b5\u03bd . The solution for \u03be\u00b5 can be shown:72.2. Linearized Gravity12\u2202\u03bd(h\u03bb\u03bb + 2\u2202\u03bb\u03be\u03bb) = \u2202\u00b5(h\u00b5\u03bd + \u2202\u00b5\u03be\u03bd + \u2202\u03bd\u03be\u00b5)\u2192 \u25a1\u03be\u03bd = 12\u2202\u03bdh\u2212 \u2202\u00b5h\u00b5\u03bd (2.6)Typically, one redefines the metric perturbation into its trace-reversedform so that h\u00b5\u03bd = h\u00b5\u03bd \u2212 12h\u03b7\u00b5\u03bd , which correspondingly obeys \u2202\u00b5h\u00b5\u03bd = 0in the de Donder gauge. This simplifies the Einstein field equations signifi-cantly, as shown in Appendix A.This gauge is useful in studying gravitational waves but also simplifiesthe form of interaction with fields. The trace-reversed form gives a straight-forward solution in terms of Green\u2019s functions of the wave equation. Forstudying interactions, such as with scalar fields, the expression\u221a\u2212gg\u00b5\u03bd canbe handled easily under integration by parts in this gauge. This form alsosignificantly simplifies the form of the graviton propagator, which is demon-strated in Appendix A.2.2.3 Transverse-Traceless GaugeThis is the gauge I will use in this thesis most frequently as it gives thegreatest simplification and combines properties of synchronous and harmonicgauges. One starts from the de Donder gauge, then applies a new conditionthat h = 0. Furthermore, the trace-reversed solution is the same as themetric perturbation; h\u00b5\u03bd = h\u00b5\u03bd . One can also set a condition n\u00b5h\u00b5\u03bd , whichallows one to further restrict the solutions to be perpendicular to a unitvector n\u00b5. The set of solutions that define this gauge are:\u2202\u03bb\u03be\u03bb =12h, \u25a1\u03be\u03bd = \u2202\u00b5h\u00b5\u03bd , n\u00b5\u2202\u00b5\u03be\u03bd + n\u03bd\u2202\u03bd\u03be\u00b5 = \u2212n\u00b5h\u00b5\u03bd (2.7)Given that the first and third are first-order differential equations that fixone degree of freedom, and the second is a second-order differential equationthat requires boundary conditions to fix \u03be\u03bd , this is a valid gauge. Settingn\u00b5 = (1, 0, 0, 0) brings back the synchronous gauge. This gauge gives aparticularly simple form of the Einstein-Hilbert action as shown in AppendixA, and is useful when decomposing the gravitational field into polarizedscalar components as described by Ford and Parker [46].2.2.4 Issues with Linearized GravityLinearized gravity will be useful in studying semiclassical and stochasticgravity, and both can be formulated as approximations using linearized82.3. Quantum Field Theorygravity. Despite this, the self-interaction and non-perturbative aspects ofgravity are not well-described by linearized gravity. As an example, theSchwartzchild metric can be written as a Taylor expansion which can be agood approximation far from a black hole. Closer to the event horizon, theapproximation becomes worse as higher-order terms become relevant. Incases where blackholes are studied, the Schwartzchild metric is taken to bethe background with metric perturbations alongside it [6].If quantum corrections are ignored, the linearized gravity approach canstill make many useful predictions. It has been used in studying cosmology[47], semiclassical gravity [48], and stochastic gravity [28, 29]. Despite this, anon-perturbative approach is often more useful and can describe phenomenanot easily accessible in linearized gravity, such as particle creation [11\u201313].Despite these issues, I will be using a linearized gravity approach frequentlythroughout this thesis.2.3 Quantum Field TheoryThe basis of our modern understanding of the building blocks of the universeis Quantum Mechanics, with the form underlying much of modern physicsbeing Quantum Field Theory (QFT). QFT can be used to describe matterand the forces that hold them together on the subatomic scale. The mostsignificant result of which is the Standard Model of Particle Physics, whichdescribes the matter particles as leptons and quarks, while the forces aredescribed as gauge bosons. This theory is adequate to explain the matterthe universe is made of and the three fundamental forces that hold themtogether, and is backed up by the particle physics experiments up to today.The basis for most of the quantum field theory done in particle physicsis to assume that the universe is filled with fields on a Minkowski space-time background. The fields are described by the classical actions of fields,which then give rise to equations of motion. Using QFT allows one to findquantum corrections to these field theories based on loop expansions, regu-larization and renormalization. These are used to formulate a Hilbert spacein which quantum mechanics can be well-defined. In particular, observablescan be calculated by finding the expectation values of operators relating toexperimental data.There are several different formalisms for calculating with QFT. As anexample, one can look at a simple example of a non-interacting massive realscalar field in Minkowski spacetime:92.3. Quantum Field TheoryL\u03d5 = 12\u03b7\u00b5\u03bd\u2202\u00b5\u03d5\u2202\u03bd\u03d5\u2212 12m2\u03d52 (2.8)The Klein-Gordon equation comes directly from this, for example viathe Euler-Lagrange equation:\u2202\u00b5L\u03d5\u2202(\u2202\u00b5\u03d5)\u2212 \u2202L\u03d5\u2202\u03d5= \u03b7\u00b5\u03bd\u2202\u00b5\u2202\u03bd\u03d5+m2\u03d5 = (\u25a1+m2)\u03d5 = 0 (2.9)A general solution to this is often given in terms of ladder operators:\u03d5(x) =\u222bd3k(2\u03c0)31\u221a2\u03c9k\u20d7(ak\u20d7+ a\u2020k\u20d7) eikx, wk =\u221ak\u20d72 +m2 (2.10)In this form, there are ladder operators for the momentum modes of k\u20d7which are plane waves eikx = e\u2212i\u03c9teik\u20d7\u00b7x\u20d7. This is a non-interacting theory,and so describes the behavior of the modes of the fields without taking intoconsideration interactions with itself or other fields.The formalism that has laid the foundation for much of Quantum FieldTheory, and which I will focus on the most is the functional formalismbased on the Feynman Path Integral formulation. This has been used mostnotably in QED and forms the basis of calculation in much of standardparticle physics. Applying this formalism to a scalar field as before exceptwith a self-interaction term:L\u03d5 = 12\u03b7\u00b5\u03bd\u2202\u00b5\u03d5\u2202\u03bd\u03d5\u2212 12m2\u03d52 +\u03bb4!\u03d54 (2.11)Functional formalism is defined by the expression:Z = \u27e8\u2126|\u2126\u27e9 =\u222bD\u03d5 eiS\u03d5+i\u222bd4x J(x)\u03d5(x) (2.12)Where S\u03d5 is the classical action for the field being quantized and J(x) is aclassical source linearly coupled to the field. The integral is over all possibleconfigurations of the field, and the result is a superposition of all the differentconfigurations as described by the Feynman path integral. This is used asa generating functional where the fields are brought down by functionalderivatives with the classical source, which is then set to zero assuming nosources. The ket |\u2126\u27e9 refers to the interacting theory ground state - whichin general is different from the free-theory ground state |0\u27e9. If taking anexpectation value where the action is the free action, which can be done by102.3. Quantum Field Theoryincluding the interacting part in the operator, the free-theory ground statecan be used:\u27e8\u2126| T O\u02c6 |\u2126\u27e9 =\u222bD\u03d5 eiS\u03d5 O\u02c6 =\u222bD\u03d5 eiS(0)\u03d5 O\u02c6\u2032 = \u27e80| T O\u02c6\u2032 |0\u27e9 (2.13)This distinction is crucial to make as the approach I will use frequentlymakes use of the free-theory ground state in perturbative expansions.As an example, if a time-ordered two-point correlation function of thefield \u03d5 is required, it can be found with the procedure:\u27e8\u2126| T \u03d5(x)\u03d5(y) |\u2126\u27e9 = ( 1i2\u03b4\u03b4J(x)\u03b4\u03b4J(y)Z)\u2223\u2223\u2223\u2223J=0=\u222bD\u03d5 eiS\u03d5\u03d5(x)\u03d5(y) (2.14)I have laid this out as a formal procedure, and I will use this frequentlythroughout this thesis. The way I have shown it derived here is formal, asto demonstrate how the procedure works from first principles, though I willfrequently use the result from (2.14) as a shorthand for the formal procedure.This procedure is useful for the interacting theory in perturbation theory.For such an expansion, one can separate out the interaction part of theaction and Taylor expand it to give an exact expression that can be evaluatedperturbatively:S\u03d5 = S(0)\u03d5 + S(I)\u03d5 , S(I)\u03d5 =\u222bd4x\u03bb4!\u03d5(x)4 (2.15)\u222bD\u03d5 eiS(0)\u03d5 eiS(I)\u03d5 \u03d5(x)\u03d5(y) =\u222bD\u03d5 eiS(0)\u03d5\u221e\u2211n=0(iS(I)\u03d5 )nn!\u03d5(x)\u03d5(y) (2.16)\u27e8\u2126| T \u03d5(x)\u03d5(y) |\u2126\u27e9 = \u27e80| T \u03d5(x)\u03d5(y) |0\u27e9+ i\u03bb24\u222bd4z \u27e80| T \u03d5(x)\u03d5(y)\u03d5(z)4 |0\u27e9\u2212 \u03bb2576\u222bd4z d4w \u27e80| T \u03d5(x)\u03d5(y)\u03d5(z)4\u03d5(w)4 |0\u27e9+O(\u03bb3)(2.17)The free theory has the property that all odd correlation functions will goto zero, and even correlation functions are broken up into all possible com-binations of two-point correlation functions according to Wick\u2019s theorem. A112.4. Regularizationnumber of terms from the expansion of the n-point correlation function canbe evaluated with Wick\u2019s theorem for each of the terms in the perturbativeexpansion.For a standard interacting quantum field theory like the \u03d54 theory, thisprocedure is straightforward and can lead one to regularization, counter-terms, and renormalization to give a simple, consistent theory. The renor-malizability of this theory is straightforward to check from the Renormal-ization Group. Only flat spacetime has been considered here, and one maywant to consider how quantum field theory can be generalized to curvedspacetimes.2.4 RegularizationIn the last section, I made clear the importance of two-point correlationfunctions in free theory as a basis for perturbative theory. The two-pointcorrelation function is also known as a propagator and is often interpretedas the probability amplitude for a particle at a coordinate x to move to acoordinate y. When there is a free momentum, such as in a free propagatoror loop, it will be integrated over. Without regularization, this leads to ul-traviolet divergence. To calculate integrals that involve propagators, such asloop regularizations of the stress-energy tensor, a UV regularization methodmust be used. The main methods are included here.\u2022 Dimensional Regularization - This is a very popular method due tothe straightforward approach. The approach is purely mathematicalrather than introducing new physics, as the dimension is analyticallycontinued. Regularizing an integral that diverges in d = 4 dimensionsis taken to d = 4 \u2212 \u03f5 and the divergences for \u03f5 \u2192 0 are found andeliminated by introducing counter-terms in the action. This methodmaintains Lorentz invariance.\u2022 Zeta Regularization - This is similar to dimensional regularizationin terms of using analytic continuation of infinite sums to convergentvalues using the Riemann zeta function in order to eliminate diver-gences.\u2022 Point Separation Regularization - This method is a standardprocedure in QFT in curved spacetime, particularly for the regular-ization of the stress-energy tensor. When finding the stress-energytensor, one may get propagators between the fields at the same point.122.4. RegularizationTo avoid divergences from this, the two points are separated and theterm that diverges as they approach is removed. This will not be usedin this thesis, though it may be another Lorent-invariant approach toconsider.\u2022 High-Energy cutoff - This is a common approach where an upperlimit \u039b is given on the magnitude of the momentum. This methodbreaks Lorentz-Invariance as it requires a special frame to have zeromomentum. This will separate out terms that rely on the cutoff sothey can be removed in the limit as \u039b \u2192 \u221e or alternatively one canstudy theories with finite cutoffs.\u2022 Pauli-Villars Regularization - This approach is one of the primaryfocuses of this thesis. The simplest form involves replacing all propa-gators with the original minus a propagator with a high mass, whichis the cutoff \u039b in this case. At low energy, the high-mass contri-bution is negligible, but at high momentum, the two terms becomeasymptotically equivalent and so the integral converges. This methodmaintains Lorentz-invariance, though otherwise includes lots of com-plications in defining the regularization in a consistent way. One mayallow for the possibility of physical regulation through some physicalcorrespondence to the \u2019ghost fields\u2019 it introduces, though this comeswith many new problems.These regularization techniques are used in different domains. Dimen-sional regularization is widespread in the study of QFT in flat spacetimeand is useful in studying renormalization and the standard model. Zeta reg-ularization is used in some similar cases such as studying the Casimir effect[49] and regularizing heat kernel approaches to QFT [50, 51].The high-energy cutoff and Pauli-Villars regularization procedures bothinvolve a cutoff. This separates it from the other methods in that one maybe able to physically interpret the cutoff. A discrete spacetime will involvea cutoff in the momentum as it sets a lower bound on the wavelength.This gives a physical interpretation to the cutoff which may give physicalconsequences. A similar thing happens with Pauli-Villars, except the cutoffis not a property of the spacetime, but the cancellation of fields. While aphysical cutoff corresponds to some extra physical scale \u039b, an extra physicalscale \u00b5 still appears in other regularization techniques such as dimensionalregularization.132.5. Quantum Field Theory in Curved Spacetime2.5 Quantum Field Theory in Curved SpacetimeAs a stepping stone to Semiclassical gravity, I will briefly outline QFT incurved spacetime (QFTCST). QFTCST consisted of many of the early ef-forts to combine quantum mechanics and general relativity by describingquantum fields on particular curved backgrounds. By expanding the fieldsin terms of Bogolubov coefficients [52]:\u03d5(x) =\u2211k[\u03b1kuk(x) + \u03b1\u2020ku\u2217k(x)] (2.18)The functions uk(x) give a complete set of solutions defined by differentmodes [52]. In a particular spacetime, the solutions may be given by differentcoefficients. The particle number is given by |\u03b1k|2, and by finding thatthis changes between backgrounds, one can determine that particle creationoccurs. Some of the more significant examples include the prediction of theHawking effect [11] and the Unruh effect [12]. In these cases, event horizonsare formed that cut off certain modes of the fields. This results in a thermalspectrum of particles related to the properties of the background, such asthe mass of the black hole or the magnitude of acceleration, to determinethe temperature [11, 12]:TH =\u210fc38\u03c0GMkB, TU =\u210fa2\u03c0ckB(2.19)This framework is inadequate when considering backreaction. By pick-ing a background, properties of the fields can be studied, but there is noprocedure for considering how those properties affect the background. Gen-eral relativity describes not just the effects of spacetime on matter, but theway matter creates that curvature. To solve more complex problems involv-ing the effects of the matter fields on spacetime, a new framework neededto be introduced.2.6 Semiclassical GravityTo describe the backreaction of quantum fields, semiclassical gravity wasdeveloped. Semiclassical gravity allows the use of the Einstein field equationsto describe the interaction of quantum matter with spacetime by taking the\u2019average\u2019 of the stress-energy tensor - ie. the expectation value. This allowskeeping the curvature classical while giving an approximation such that the142.6. Semiclassical Gravityquantum matter is described like classical matter. This method eliminatesquantum fluctuations but is useful for many purposes.Given some quantum matter, which I will take to be a massive scalarfield:S\u03d5 =\u222bd4x12\u221a\u2212g(x) (g\u00b5\u03bd(x)\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u2212m2\u03d5(x)2) (2.20)There are several ways to approach calculating the stress-energy tensor.In the most straightforward approach, the formula is given by:\u27e8T\u03b1\u03b2(x)\u27e9 = 2\u221a\u2212g(x) \u27e8 \u03b4S\u03d5\u03b4g\u03b1\u03b2(x)\u27e9=12g\u03b1\u03b2(x)(g\u00b5\u03bd(x) \u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9 \u2212m2 \u27e8\u03d5(x)2\u27e9)\u2212 g\u03b1\u00b5(x)g\u03b2\u03bd(x) \u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9(2.21)In this expression, the expectation values are found after simply takingthe variation with respect to the metric. This can be done in the functionalformalism by taking the expression:\u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9 = Z\u22121\u222bD\u03d5 eiS\u03d5\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x) (2.22)Here, Z\u22121 must be included as a normalization, where Z is the generatingfunctional as defined in (2.12).There is another way to formulate this which lines up better with theinterpretation given by the functional approach. Assuming the action willtake the form of an effective action Seff [g] = SEH [g] + \u0393[g], where \u0393[g] isan effective potential found by integrating over the fields:ei\u0393[g] =\u222bD\u03d5 eiS\u03d5[\u03d5,g] (2.23)When \u0393 is used as the action in this case, the normalization naturallycomes out of taking the variation:\u03b4\u0393\u03b4g\u00b5\u03bd(x)= \u2212ie\u2212i\u0393\u222bD\u03d5 eiS\u03d5(i \u03b4S\u03d5\u03b4g\u00b5\u03bd(x)) (2.24)This is how I will define the semiclassical approach as it has several ben-efits. Instead of the expectation value being added afterward, the procedure152.6. Semiclassical Gravitygives the expectation value of the quantum stress-energy tensor operator,including the normalization. The gravitational perturbations do not need tobe treated as quantum; it is never integrated out, and no quantum effects orpropagators for the graviton are considered, though it is possible to includethese if desired. It also allows a non-perturbative approach in some cases,and can be mixed with classical approaches to studying gravity, particularlyin the study of cosmology.Given that this is the interaction of two systems, a quantum systemdescribed by \u03d5 and an environment described by the gravitational field, thisapproach is similar to the Feynman-Vernon influence functional formalism[32]. The matter field degrees of freedom are integrated out, leaving aneffective action for the gravitational field. The effective action \u0393 is sometimescalled the influence action, and the generating functional ei\u0393 is called theinfluence functional [6].2.6.1 Perturbative Semiclassical GravityFollowing the approach of Pasukonis [48], one can use a perturbative expan-sion of the metric. If one decomposes the metric into a background and aperturbation g\u00b5\u03bd(x) = b\u00b5\u03bd(x) + h\u00b5\u03bd(x), one can expand the action aroundthe background. I will use the Minkowski metric as the background in thiswork as the main focus will be on the Minkowski vacuum. In Appendix A,I have given tensors F and G which can be used to write expansions aroundthe Minkowski metric.In this case, the real massive scalar field action can be expanded as:S\u03d5 = S(0)\u03d5 + S(1)\u03d5 + S(2)\u03d5 + ...=\u222bd4x (12\u03b7\u00b5\u03bd\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u2212 12m2\u03d5(x)2)+\u222bd4x (F\u00b5\u03bd\u03b1\u03b2\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u2212 12F\u03b1\u03b2m2\u03d5(x)2)h\u03b1\u03b2(x)+12\u222bd4x (F\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u2212 12F\u03b1\u03b2\u03c3\u03c1m2\u03d5(x)2)h\u03b1\u03b2(x)h\u03c3\u03c1(x) +O(h3)(2.25)From this point, the standard tools of perturbative QFT are available.First, a free theory to expand around should be found - in this case, the firstterm in the expansion is the free scalar field in Minkowski spacetime. Fromthis, the standard propagators can be derived. All of the remaining terms162.6. Semiclassical Gravityare interaction terms between two scalar fields and a number of perturbedmetric terms. If h\u03b1\u03b2(x) is small, the expansion can be approximated bycutting off at a certain order. This can be made more explicit by writingthe expansion in terms of \u03ba [48], though I will just maintain that h is smallfor the purposes here.The variation of the effective potential can be written as an expansion.To zeroth order in h\u03b1\u03b2(x):\u03b4\u0393\u03b4h\u03b1\u03b2(x)= \u2212ie\u2212i\u0393\u222bD\u03d5 eiS\u03d5(i \u03b4S\u03d5\u03b4h\u03b1\u03b2(x)) \u223c e\u2212i\u0393\u222bD\u03d5 eiS(0)\u03d5 \u03b4S(1)\u03d5\u03b4h\u03b1\u03b2(x)= F\u00b5\u03bd\u03b1\u03b2 \u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9 \u2212 12F\u03b1\u03b2m2 \u27e8\u03d5(x)2\u27e9(2.26)Here, the expectation values will be the propagators in the free theory.Substituting in for the F tensors and comparing to the non-perturbativeresult in the Minkowski metric will show they are equivalent:\u27e8T\u03b1\u03b2(x)\u27e9 = 2\u221a\u2212g(x) \u27e8 \u03b4S\u03d5\u03b4g\u03b1\u03b2(x)\u27e9\u223c 12\u03b7\u03b1\u03b2(\u03b7\u00b5\u03bd \u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9 \u2212m2 \u27e8\u03d5(x)2\u27e9)\u2212 \u03b7\u03b1\u00b5\u03b7\u03b2\u03bd \u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9= 2F\u00b5\u03bd\u03b1\u03b2 \u27e8\u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)\u27e9 \u2212 F\u03b1\u03b2m2 \u27e8\u03d5(x)2\u27e9(2.27)This demonstrates that the stress-energy tensor in flat spacetime is de-pendent on these correlation functions at the point x. This should be in-variant under changes in x, and so this will be a constant at every point inspacetime. If this is found to be non-zero in the vacuum state, this corre-sponds to a non-zero vacuum energy density.Regularizing using high-energy cutoff, one can take the field operators interms of ladder operators. Assuming a massless field as an approximationwhen \u039b >> m:\u27e80| \u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x) |0\u27e9 =\u222bd3k d3l(2\u03c0)6(ik\u00b5)(il\u03bd)2\u221a\u03c9k\u03c9l\u27e80| ala\u2020\u2212k |0\u27e9 ei(k+l)x=\u222bd3k d3l(2\u03c0)6\u2212k\u00b5l\u03bd2\u221a\u03c9k\u03c9l(2\u03c0)3\u03b4(k + l)ei(k+l)x=\u222bd3k(2\u03c0)3k\u00b5k\u03bd2\u03c9k(2.28)172.6. Semiclassical GravitySetting m to zero, which is approximately valid for a high cutoff, gives\u03c9k =\u2223\u2223\u2223\u20d7k\u2223\u2223\u2223. In the case that Lorentz invariance is broken, there will be aseparation of the time and space components of k\u00b5. The spatial componentswill just be given by k0 = \u03c9k =\u2223\u2223\u2223\u20d7k\u2223\u2223\u2223, while the spatial components can bewritten in cartesian coordinates, which makes it clear that averaging overwill give 13 for each component. When integrating the solid angle, onlydiagonal terms in the spatial components will be non-zero as the off-diagonalterms cancel due to symmetry. The spatial terms will get a factor of 13 , so Iwill define a metric W\u00b5\u03bd to write this as:\u27e80| \u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x) |0\u27e9 = \u039b416\u03c02W\u00b5\u03bd , W\u00b5\u03bd =\uf8eb\uf8ec\uf8ec\uf8ed1 0 0 00 13 0 00 0 13 00 0 0 13\uf8f6\uf8f7\uf8f7\uf8f8 (2.29)This can be written as T\u03b1\u03b2 = T0W\u00b5\u03bd , where T0 =\u039b416\u03c02.2.6.2 Next-order ContributionIt will be useful to look at the second derivative of the effective action tosee how the assumptions of semiclassical gravity behave at this order:\u03b42\u0393\u03b4h\u03b1\u03b2(x)\u03b4h\u03c3\u03c1(y)=\u03b4\u03b4h\u03c3\u03c1(y)[e\u2212i\u0393\u222bD\u03d5 eiS\u03d5( \u03b4S\u03d5\u03b4h\u03b1\u03b2(x))]= \u2212ie\u22122i\u0393[\u222bD\u03d5 eiS\u03d5( \u03b4S\u03d5\u03b4h\u03b1\u03b2(x))][\u222bD\u03d5 eiS\u03d5( \u03b4S\u03d5\u03b4h\u03c3\u03c1(y))]+ ie\u2212i\u0393\u222bD\u03d5 eiS\u03d5( \u03b4S\u03d5\u03b4h\u03b1\u03b2(x))(\u03b4S\u03d5\u03b4h\u03c3\u03c1(y)) + e\u2212i\u0393\u222bD\u03d5 eiS\u03d5( \u03b42S\u03d5\u03b4h\u03b1\u03b2(x)\u03b4h\u03c3\u03c1(y))(2.30)The normalization remains, and the first and second terms are imaginary.The third term is real, though, in the standard perturbative free theory, itwill only be non-zero when x = y. Imaginary terms make no sense for thestress-energy tensor, which as an observable should be real. The issue is aresult of defining the expectation value with respect to the vacuum.To interpret this, one has to consider the nature of the vacuum. Thiscan be done by defining an \u2019in\u2019 vacuum |0\u2212\u27e9 at early times and an \u2019out\u2019vacuum |0+\u27e9 at late times. This means that the expectation value can bewritten by redefining the generating function Z [6]:182.7. CosmologyZ = \u27e80+|0\u2212\u27e9 = ei\u0393, e\u2212i\u0393\u222bD\u03d5 eiS\u03d5( \u03b4S\u03d5\u03b4h\u03b1\u03b2(x)) =\u27e80+| \u03b4S\u03d5\u03b4h\u03b1\u03b2(x) |0\u2212\u27e9\u27e80+|0\u2212\u27e9 (2.31)If the effective action \u0393 is imaginary, as would be suggested by the pre-vious analysis, it can be found that the in and out states are not equal[6]:|\u27e80+|0\u2212\u27e9|2 = ei\u0393e\u2212i\u0393\u2217 = e\u22122 Im(\u0393) \u223c 1\u2212 2 Im(\u0393) (2.32)If these were the same states, they would be normalized to one. The re-maining term P = 2 Im(\u0393) corresponds to a probability of particle creation.The stress-energy tensor can be written in terms of the effective actionin a form that makes it clear how it will behave in a slightly perturbedspacetime by writing it as a Taylor expansion:\u0393 =\u222bd4y\u03b4\u0393\u03b4h\u03b1\u03b2(y)h\u03b1\u03b2(y)+12\u222bd4yd4z\u03b42\u0393\u03b4h\u03b1\u03b2(y)\u03b4h\u03c3\u03c1(z)h\u03b1\u03b2(y)h\u03c3\u03c1(z)+O(h3)(2.33)\u27e8T\u00b5\u03bd(x)\u27e9 \u223c 2 \u03b4\u0393\u03b4h\u00b5\u03bd(x)= 2\u03b4\u0393\u03b4h\u00b5\u03bd(x)+\u222bd4z\u03b42\u0393\u03b4h\u00b5\u03bd(x)\u03b4h\u03c3\u03c1(z)h\u03c3\u03c1(z) + ...(2.34)It is clear that in a flat spacetime, only the zeroth order term will remain,which corresponds to the expectation value calculated before. In a perturbedspacetime, there can be extra terms that may be real and correspond to aperturbation in the stress-energy tensor, or imaginary and correspond toparticle creation.2.7 CosmologyThe main area in which semiclassical gravity has made observable predic-tions is cosmology. Modern cosmological models such as \u03bbCDM assume theuniverse can be described on large scales by an approximately homogeneousand isotropic metric - the Friedmann-Robertson-Walker (FRW) metric:ds2 = \u2212dt2 + a2(t)( dr21\u2212 kr2 + r2d\u21262) (2.35)192.7. CosmologySolving the Einstein field equations with this metric gives the Fried-mann equations which describe the relationship between the different formsof energy in the universe and the corresponding large-scale structure andevolution over time as described by a(t). By finding the Ricci tensor for thismetric, it can be substituted into the Einstein field equations.Also following from the ideas of homogeneity and isotropy, the contents ofthe universe are assumed to be perfect fluids studied in comoving coordinatesU\u00b5 = (1, 0, 0, 0). The stress-energy tensor for a relativistic perfect fluid isthen given by:T\u00b5\u03bd = (\u03c1+ p)U\u00b5U\u03bd + pg\u00b5\u03bd (2.36)Here, p is pressure and \u03c1 is energy density. Substituting in for U\u00b5 andthe FRW metric gives:T\u00b5\u03bd =\uf8eb\uf8ec\uf8ec\uf8ed\u03c1 0 0 00 a2p 0 00 0 a2p 00 0 0 a2p\uf8f6\uf8f7\uf8f7\uf8f8 (2.37)The equation of state is defined by relating the pressure and density -typically written as:p = w\u03c1 (2.38)Here, w will depend on and define the equation of state of the fluid.Matter is assumed to have minimal pressure compared to its energy densityas described by E = mc2; matter is therefore described by w = 0. Radiationhas a pressure one-third of the energy density, so w = 13 . One can make notethat the corresponding stress-energy tensor for radiation in flat spacetimewill be given by T\u00b5\u03bd = \u03c1W\u00b5\u03bd , so the form given in the previous section forthe non-relativistic vacuum energy density is in fact that of radiation. ALorentz-invariant vacuum energy density is proportional to the metric, sothat is given by w = \u22121.When considering the contents of the universe, one can consider eachcomponent to have some average energy density which each contributes sep-arately. The Friedmann equations give a relationship between the scale fac-tor and one of the components in terms of its energy density and pressure.These are, written in terms of the Hubble parameter H = a\u02d9a :H2 =8\u03c0G3\u03c1\u2212 ka2(2.39)202.7. CosmologyH\u02d9 +H2 =\u22124\u03c0G3(\u03c1+ 3p) (2.40)The equation (2.39) demonstrates an important property of the curva-ture. Given a Hubble parameter, which can be determined from observation,the curvature is determined by the energy density of all of the other compo-nents. Conversely, if the curvature is determined - again, through observa-tions - the energy density of all other components must obey a relationshipto give the observed curvature.To make this clearer, a parameter \u2126i for each component is defined as:\u2126i =8\u03c0G3H2\u03c1i =\u03c1i\u03c1c(2.41)This is written as a relative fraction of each component\u2019s energy densitycompared to some critical density \u03c1c, defined as the density that gives theHubble parameter today (H0) for zero flatness today (k = 0). Each com-ponent will evolve differently, but these values can be found in the universeas it is today. The curvature can also be written in the same form \u2126k asan effective density parameter so that the first Friedmann equation can bewritten as: \u2211i\u2126i = 1, \u2126k =\u2212kH2a2(2.42)Each component evolves with the scale factor differently based on itsequation of state:\u2126i = \u2126i0a\u22123(w+1) (2.43)This gives a\u22124 dependence for radiation, a\u22123 for matter, and no depen-dence for vacuum energy, which means that while radiation and matter getdiluted over time, vacuum energy does not. Curvature also varies with a\u22122,which is one of the motivations behind introducing inflation, given that arapid expansion would remove the preexisting curvature [17]. This givesa compact form for the Hubble parameter which evolves as the universeexpands based on the components:H2 = H20 (\u2126\u03bb +\u2126k0a\u22122 +\u2126m0a\u22123 +\u2126r0a\u22124) (2.44)Where H0 is the Hubble parameter today, about 67km\/s\/Mpc [53], andthe density parameters as they are today at a = 1 for vacuum energy,curvature, matter, and radiation.212.7. CosmologyThrough observations of the CMB (particularly [53\u201355]), it has beenfound that \u2126k0 \u223c 0. Through WMAP and supernova observations [4, 53,54], it has been found that matter makes up roughly 32% of the energydensity (\u2126m0 = 0.32), while most of the remaining 68% appear to be avacuum energy (\u2126\u03bb = 0.68) with the equation of state given by w \u223c \u22121[54]. Radiation is assumed to be negligible, though it was more dominantat earlier times due to the a\u22124 dependence.2.7.1 The Cosmological Constant ProblemIn the solution of the Einstein field equations to get the Friedmann equa-tions, a cosmological constant term can be included. This can then be usedto find the energy density term that gives the \u2126\u03bb parameter. It would there-fore appear that there is a non-zero cosmological constant. With a Hubbleconstant of 67 (km\/s)\/Mpc, the energy density corresponding to the valueof \u2126\u03bb = 0.7 is about 5\u00d7 10\u221210J\/m3.Looking at the prediction from semiclassical gravity, one can see thatT00 = \u03c1 =\u039b416\u03c02. If this cutoff is at the Planck scale, this is different fromthe observed value by about a factor of 10120 [5]. This demonstrates thatthe cutoff method for determining the vacuum energy density is completelyinvalid - there must be some process by which this value is canceled. How-ever, given there is still a small apparent cosmological constant left over, analternative explanation is needed. There have been several possible solutionsproposed.2.7.2 Bare Cosmological Constant and Fine-tuningBianchi and Rovelli [24] suggest that a bare cosmological constant is a validsolution to the problem. There are many constants in physics, and havingan extra parameter that fits into the description of gravity may not be toounexpected.The arguments against this typically boil down to a fine-tuning argu-ment, which Bianchi and Rovelli attempt to explain away. The cosmiccoincidence argument against a fine-tuned cosmological constant is basedon the cosmological principle, that we do not expect our place in the uni-verse to be privileged. Given the fact that the density of matter scales witha\u22123 while the cosmological constant does not, the coincidence of us beingable to exist is based on these two densities being roughly equal to allowthe stability required. Bianchi and Rovelli argue against this based on theanthropic principle - ie. we must always observe the universe from a special222.7. Cosmologyperspective: one where we can exist.If the cosmological constant is a given, matter density in our region of theuniverse is still a variable given by the initial conditions caused by quantumfluctuations in the early universe [47]. Our existence at a time when acertain patch of matter exists to allow us to exist depends solely on thosefluctuations and an adequately large universe, and not the cosmologicalconstant.Even given a particular matter density, the cosmological constant shouldalso have some freedom while still maintaining relative stability. For thispurpose, a stable universe is one that does not expand too quickly to allowstructure to form, or contract back upon itself sooner than around 1010 years.The density required for stability is the aforementioned \u2019critical density\u2019 thatgives k = 0:\u03c1c =3H208\u03c0G,\u2211i\u2126i = 1, \u2126k = 0 (2.45)If the cosmological constant were slightly different, one could still havea stable universe so long as the Hubble parameter also changes, making itso that \u2126\u039b+\u2126m \u223c 1 is still satisfied - significant deviations from this wouldbe unstable. By assuming global curvature is zero as an initial condition(ie. due to inflation) one can still satisfy a stability condition given theseinitial conditions, and so there is some degree of freedom in the cosmologicalconstant that also sets the corresponding stable Hubble parameter. As aresult, the fine-tuning argument may not be as relevant for the cosmologicalconstant problem as is often argued.The aspect of the cosmological constant relevant to theoretical physicsis the mechanism by which it occurs. Bianchi and Rovelli argue that thereis not good reason to identify the cosmological constant with QFT, partic-ularly due to the failure of the UV cutoff method. Instead, it should beconsidered a prediction of GR with a particular value assigned. This wouldsuggest that the cosmological constant is a fine-tuned constant. A preferableinterpretation may be that, while there is some physical process involved, itis not exactly a cosmological constant. Bianchi and Rovelli give an indica-tion of a type of effect that can be derived from QFT that does have someinfluence on gravitation - changes in the vacuum.In QFT, one can calculate mass shifts in quantities such as the massesof nuclei that correspond directly with interaction with the vacuum [24].These calculations have been very successful in predicting experimental re-sults, such as masses. If these vacuum corrections had no influence on the232.7. Cosmologygravitational field, one would expect violations of the equivalence principle;inertial mass due to the corrections would differ from gravitational massfrom interaction with the gravitational field. However, as Bianchi and Rov-elli point out, these corrections are on the order of 10\u22123, while the equiv-alence principle has been shown to be accurate to at least 10\u221212 accuracy[56].Given the good reason to expect changes in the vacuum to gravitate, itmay be worth considering models where the vacuum has slight changes andcan be tuned with reasonable parameters as opposed to the fine-tuning of acosmological constant parameter.2.7.3 QuintessenceQuintessence is a semiclassical model that tries to resolve the cosmologi-cal constant problem through the introduction of a field with an evolvingequation of state. The concept is a similar concept to inflation, where the be-havior of a scalar field has a significant effect on the expansion of spacetime[17].Quintessence involves a scalar field with a relation between the densityand pressure that gives a particular type of time-dependent equation of state[26]. This means that it behaves differently from the other types of energydiscussed previously in the context of the Friedmann equations. This modelhas been shown to be a possible alternative to \u03bbCDM [57] which is valid inthe semiclassical approach.One proposal is a \u2019tracker\u2019 behavior where the time-dependence allowsit to behave similarly to radiation in the early universe with w \u223c 0.3, butevolve towards an equation-of-state more like a cosmological constant withw \u223c \u22120.8 later on [58]. This is introduced as a way to get around thecosmic coincidence problem so that the equation of state does not need tobe fine-tuned in the same way as the cosmological constant. As discussedpreviously, this problem may not need a solution, and so this requirementof quintessence serves to complicate the model.This model requires a new component in the cosmological model thatacts differently from any of the other components for which the \u03bbCDMmodelusually accounts, such as matter, radiation, curvature, and a cosmologicalconstant. There are several types of quintessence models [59] with variousparameters that can be tuned and fit to observations. Some of these modelsinvolve new types of physics, such as supersymmetry, and so quintessencemodels may be a testable cosmological application of certain particle physicsmodels. Observations so far are not able to distinguish between a \u03bbCDM and242.8. Beyond Semiclassical Gravityso-called QCDM models [59]. Many of the apparent benefits of quintessenceare predicated on the requirement to fix the fine-tuning problem by allowingevolution, though this may not be a real problem. If the model introducedrequires new physics such as a time-evolving equation of state, and assump-tions to fix problems that may not be real problems, it may be preferableto find an alternative that does not make many new assumptions instead.2.7.4 Vacuum FluctuationsAs it is suggested that changes in the vacuum can gravitate, one should notdiscount vacuum fluctuations. Semiclassical gravity is defined by discount-ing vacuum fluctuations and assuming \u27e8T\u00b5\u03bd(x)T\u03b1\u03b2(y)\u27e9 \u223c \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T\u03b1\u03b2(y)\u27e9,however this may not always be valid. If one considers that the expectationvalue as a constant is not able to interact with gravity, the vacuum fluctua-tions may be the lowest-order contribution of the vacuum to the gravitationalfield. Wang et al. [25] considered this possibility, which will be the approachI will focus on in this thesis.2.8 Beyond Semiclassical GravityIn this chapter, I have discussed some of the approaches and predictions ofSemiclassical gravity. By neglecting fluctuations and using the expectationvalue of the stress-energy tensor, one can use the methods of General Rel-ativity such as the Friedmann equations with the expectation value of thestress-energy tensor directly substituted. This has been used to solve manyproblems, however, this approximation still leaves out cases where fluctua-tions may be relevant. To go beyond semiclassical gravity, an approach thattakes into account fluctuations and other next-order contributions such asdissipation is required.25Chapter 3Stochastic GravityNote: This chapter contains an overview of stochastic gravity. A more de-tailed overview of this topic can be found primarily in the textbook [6].In the last section, I demonstrated how semiclassical gravity can be de-rived by introducing an effective potential for the quantum fields, leading toan \u2019averaging out\u2019 of the quantum fluctuations. This was adequate to getsome useful results, including the stress-energy tensor, however, it comeswith some issues. Many quantum phenomena are not explained by theirexpectation values alone. If there is a superposition of two states, the semi-classical approach will only give a result of the expectation value of the twostates. One may expect instead that the gravitational field itself shouldbe in a superposition if it is quantum. Treating the gravitational field asquantum does, however, come with some issues.3.1 Issues with Quantum GravityIn writing out the stress-energy tensor, it is clear that it is made up ofquantum field operators, those being the field operators. This implies thatthe stress-energy tensor itself can be interpreted as a quantum operator,which naturally leads to the interpretation of the curvature as also beingquantum based on the Einstein field equations. Looking at linearized gravityand the formalism I used in the previous chapter, one can see similaritiesto the quantization of fields in QFT through the introduction of quantumoperators and perturbation theory.The form of the graviton propagator and Einstein-Hilbert action in lin-earized gravity has been well studied [60\u201362]. Similar to Choi et al. [61],one can derive many quantities by combining perturbative expansions of themetric around a background spacetime - in this case, flat spacetime. Forexample, the Einstein-Hilbert action can be expanded to arbitrarily highorder, such as in (A.24).If this is interpreted as a quadratic \u2019free-theory\u2019 propagator giving thegraviton propagator in this gauge and interaction terms at h3, h4 and soon, then one can see that there are Feynman rules that can be derived263.2. Schwinger-Keldysh Formalismfrom this expansion. It is clear, however, that the coupling constant for theinteraction terms is the same, meaning that while this works fine for smallh, at large h the interaction terms will be very large. To renormalize thistheory, one would need to introduce counter-terms for each term, and thiswould be an infinite amount simply by inspection. It can be shown thatthe order of Feynman diagrams increases with loop order [1], which directlyleads to non-renormalizability. Other arguments can be made for the non-renormalizability from the nature of gravity, such as the way it behavesat high-energy compared to quantum field theories, which asymptoticallyapproach a conformal field theory [63].There are theories that have been proposed to make quantum gravitywork, such as loop quantum gravity and string theory [7, 8]. These theories,however, introduce new physics such as strings which are only testable athigh energies far outside what can be replicated in experiments.In this chapter, I will discuss one possible \u2019bottom-up\u2019 approximation toquantum gravity - Stochastic Semiclassical Gravity, or just \u2019Stochastic Grav-ity\u2019 - that, unlike the \u2019top-down\u2019 approaches of string theory, loop quantumgravity and others, may require introducing very little new physics, and maybe able to make predictions in domains where standard semiclassical gravitymay not be useful.3.2 Schwinger-Keldysh FormalismStochastic Gravity relies on a formalism related to functional formalism,called the Schwinger-Keldysh formalism [30, 31]. In the formalism I haveused so far, correlation functions are calculated between in and out vacua,that is to say, that the \u2019out\u2019 vacuum occurs at an asymptotically free stateat t \u2192 +\u221e, while the \u2019in\u2019 vacuum is asymptotically free at t \u2192 \u2212\u221e. Ifthere are gravitational perturbations between these two times then the \u2019out\u2019vacuum may not be equivalent to the \u2019in\u2019 vacuum. This can lead to complexterms in the expectation value of the stress-energy tensor calculated in the\u2019In-Out\u2019 formalism which corresponds to particle creation. This may beuseful in calculating certain effects, however, the physical expectation valueof the stress-energy tensor should be real [6]. The Schwinger-Keldysh \u2019In-In\u2019formalism allows this.The idea behind this formalism is to introduce two copies of each field,the metric, and classical sources which are ordered forwards and backwardin time. These quantities are then made to line up at the specific time thatis being looked at so that there is a \u2019Closed Time Path\u2019 from the in-vacuum273.2. Schwinger-Keldysh Formalismat ti to the time tf and then back to itself. The generating functional Z inthis formalism takes the form (10.33 in [6]):Z =\u222bD\u03d5+D\u03d5\u2212 \u03b4(\u03d5+(tf )\u2212 \u03d5\u2212(tf ))exp(i\u222b tfti\u222bd3x (L[\u03d5+, h+] + J+\u03d5+) + i\u222b titf\u222bd3x (L[\u03d5\u2212, h\u2212] + J\u2212\u03d5\u2212))(3.1)Formally, this is defined by replacing the mass factor m2 in the actionfor \u03d5 with m2 \u2212 i\u03f5 for the action for \u03d5+ to give the boundary conditions,then taking the action to be iS\u03d5[\u03d5+, h+] + (iS\u03d5[\u03d5\u2212, h\u2212])\u2020 [6]. This allowsthe boundary condition for the negative fields to be modified and gives thereversed time-ordering in the integration.This form will introduce a minus sign between the positive and nega-tive time-direction terms in the action. The sources allow for correlationfunctions to be calculated between the different fields, and varying with re-spect to the metric allows one to calculate the expectation values of thestress-energy tensor. It is straightforward to see that, to the lowest order,this reproduces the result from semiclassical gravity. The difference betweentaking the variation using the positive and negative fields is a negative sign.The Schwinger-Keldysh versions of the effective potential and stress-energytensor are given by (10.36 in [6]):ei\u0393 =\u222bD\u03d5+D\u03d5\u2212 \u03b4(\u03d5+(tf )\u2212 \u03d5\u2212(tf )) exp(iS+ \u2212 iS\u2212) (3.2)\u27e8T\u00b5\u03bd(x)\u27e9 = 2\u221a\u2212g+(x) \u03b4\u0393\u03b4g+\u00b5\u03bd(x)\u2223\u2223\u2223\u2223h\u00b1=h=\u22122\u221a\u2212g\u2212(x) \u03b4\u0393\u03b4g\u2212\u00b5\u03bd(x)\u2223\u2223\u2223\u2223h\u00b1=h(3.3)The form of the Lagrangian is the same in both cases, except substitutingthe positive or negative versions of the fields:L[\u03d5\u00b1(x), h\u00b1] = 12\u03b7\u03b1\u03b2\u2202\u03b1\u03d5\u00b1(x)\u2202\u03b2\u03d5\u00b1(x)\u2212 12m2\u03d5\u00b1(x)2+ F\u03b1\u03b2\u03c3\u03c1 \u2202\u03b1\u03d5\u00b1(x)\u2202\u03b2\u03d5\u00b1(x)h\u00b1\u03c3\u03c1(x)\u221212F\u03c3\u03c1m2\u03d5\u00b1(x)2 h\u00b1\u03c3\u03c1(x) +O(h2)(3.4)Finally, one can demonstrate that in the lowest order, one gets back thesame result as in the semiclassical case, except by introducing a minus signwhen taking the variation with respect to the negative metric:283.3. Stochastic Gravity from Schwinger-Keldysh Functional Formalism\u03b4\u0393\u03b4g+\u00b5\u03bd(x)= \u2212ie\u2212i\u0393\u222bD\u03d5+D\u03d5\u2212 exp(iS+ \u2212 iS\u2212)i \u03b4S+\u03b4h+\u00b5\u03bd(x)\u2248 e\u2212i\u0393\u222bD\u03d5+ exp(iS(0)+)(F\u03b1\u03b2\u00b5\u03bd \u2202\u03b1\u03d5+(x)\u2202\u03b2\u03d5+(x)\u2212 12F\u00b5\u03bdm2\u03d5+(x)2)= F\u03b1\u03b2\u00b5\u03bd \u27e8\u2202\u03b1\u03d5(x)\u2202\u03b2\u03d5(x)\u27e9 \u2212 12F\u00b5\u03bdm2 \u27e8\u03d5(x)2\u27e9(3.5)\u03b4\u0393\u03b4g\u2212\u00b5\u03bd(x)= \u2212ie\u2212i\u0393\u222bD\u03d5+D\u03d5\u2212 exp(iS+ \u2212 iS\u2212)i \u03b4S\u2212\u03b4h\u2212\u00b5\u03bd(x)\u2248 \u2212e\u2212i\u0393\u222bD\u03d5\u2212 exp(\u2212iS(0)\u2212)(F\u03b1\u03b2\u00b5\u03bd \u2202\u03b1\u03d5\u2212(x)\u2202\u03b2\u03d5\u2212(x)\u2212 12F\u00b5\u03bdm2\u03d5\u2212(x)2)= \u2212F\u03b1\u03b2\u00b5\u03bd \u27e8\u2202\u03b1\u03d5(x)\u2202\u03b2\u03d5(x)\u27e9+ 12F\u00b5\u03bdm2 \u27e8\u03d5(x)2\u27e9(3.6)Given this result, it seems that the semiclassical results can still be ob-tained in this formalism. While this method is slightly different, there donot appear to be any differences to the physics involved. The Lagrangianused is of the same form, the only difference being that the closed time pathis taken forwards and backward with this Lagrangian. This should not in-troduce any new physical concepts as the backward evolution of the systemshould also be given by the same Lagrangian under time-reversal symmetry.This formalism, along with the Feynman-Vernon influence action [32] arethe only new concepts that must be introduced to give stochastic gravity.3.3 Stochastic Gravity from Schwinger-KeldyshFunctional FormalismTo derive stochastic gravity, the second derivative of the effective potential istaken. Once that is done, the effective action can be written in a new Taylor-expanded form in order to get back each term from taking derivatives of theeffective action. The reason this formalism allows this is the separationbetween the positive and negative fields exists for the purpose of findingderivatives but vanishes in the limit as they approach a common value. Thisis different from the results in semiclassical gravity because the imaginaryterms in that case corresponded to a change in the vacuum - ie. particlecreation.293.3. Stochastic Gravity from Schwinger-Keldysh Functional FormalismFollowing on from the previous section, the second-order derivatives canbe taken. The result looks similar at first, but by taking the positive andnegative field derivatives, a subtle addition appears:\u03b42\u0393\u03b4g+\u00b5\u03bd(x)\u03b4g+\u03b3\u03f5(y)=\u03b4\u03b4h+\u03b3\u03f5(y)[e\u2212i\u0393\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u00b5\u03bd(x))]= \u2212ie\u22122i\u0393[\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u00b5\u03bd(x))][\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u03b3\u03f5(y))]+ ie\u2212i\u0393\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u00b5\u03bd(x))(\u03b4S+\u03d5\u03b4h+\u03b3\u03f5(y))+ e\u2212i\u0393\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b42S+\u03d5\u03b4h+\u00b5\u03bd(x)\u03b4h+\u03b3\u03f5(y))(3.7)Some care must be taken for the ordering of the operators. In semi-classical gravity, the functional integrals already implicitly include the timeordering, but in this formalism, it is slightly more complicated. Taking thecorrelation function of two positive fields will give time ordering, and twonegative fields will give reverse time ordering. Examining this, first one cantake the limit as h+ \u2192 h and substitute in the stress-energy tensors:4\u221a\u2212g(x)\u221a\u2212g(y) \u03b42\u0393\u03b4g+\u00b5\u03bd(x)\u03b4g+\u03b3\u03f5(y)= \u2212i \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T \u03b3\u03f5(y)\u27e9+ i \u27e8T (T\u00b5\u03bd(x)T \u03b3\u03f5(y))\u27e9+4\u221a\u2212g(x)\u221a\u2212g(y) \u27e8\u03b4T\u00b5\u03bd(x)\u03b4h\u03b3\u03f5(y) \u27e9(3.8)The expectation value in the second term on the right can be split intoreal and imaginary parts. I will define these in the same way as Hu andVerdaguer (10.38 in [6]):\u2212i \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T \u03b3\u03f5(y)\u27e9+ i \u27e8T (T\u00b5\u03bd(x)T \u03b3\u03f5(y))\u27e9 = iN\u00b5\u03bd\u03b3\u03f5(x, y)\u2212H\u00b5\u03bd\u03b3\u03f5S (x, y)(3.9)N\u00b5\u03bd\u03b3\u03f5(x, y) = Re \u27e8T (T\u00b5\u03bd(x)T \u03b3\u03f5(y))\u27e9 \u2212 \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T \u03b3\u03f5(y)\u27e9 (3.10)H\u00b5\u03bd\u03b3\u03f5S (x, y) = Im \u27e8T (T\u00b5\u03bd(x)T \u03b3\u03f5(y))\u27e9 (3.11)303.3. Stochastic Gravity from Schwinger-Keldysh Functional FormalismK\u00b5\u03bd\u03b3\u03f5(x, y) =\u22124\u221a\u2212g(x)\u221a\u2212g(y) \u27e8\u03b4T\u00b5\u03bd(x)\u03b4h\u03b3\u03f5(y) \u27e9 (3.12)4\u221a\u2212g(x)\u221a\u2212g(y) \u03b42\u0393\u03b4g+\u00b5\u03bd(x)\u03b4g+\u03b3\u03f5(y) = iN\u00b5\u03bd\u03b3\u03f5(x, y)\u2212H\u00b5\u03bd\u03b3\u03f5S (x, y)\u2212K\u00b5\u03bd\u03b3\u03f5(x, y)(3.13)To get the real and imaginary parts, one can see how each ordering splitsinto symmetric and antisymmetric parts. Using the anticommutator {, } andcommutator [, ] one can write each ordering:\u27e8T\u00b5\u03bd(x)T \u03b3\u03f5(y)\u27e9 = 12\u27e8{T\u00b5\u03bd(x), T \u03b3\u03f5(y)}\u27e9+ 12\u27e8[T\u00b5\u03bd(x), T \u03b3\u03f5(y)]\u27e9 (3.14)\u27e8T \u03b3\u03f5(y)T\u00b5\u03bd(x)\u27e9 = 12\u27e8{T \u03b3\u03f5(y), T\u00b5\u03bd(x)}\u27e9+ 12\u27e8[T \u03b3\u03f5(y), T\u00b5\u03bd(x)]\u27e9 (3.15)It is clear that the symmetrized term in both will be independent ofordering, so the time ordering makes no difference to the N kernel. Further-more, given that the stress-energy tensor is self-adjoint, one can show thatit is also real and positive semi-definite (pg. 342-343 in [6]):(\u27e8\u03c8| {T\u00b5\u03bd(x), T \u03b3\u03f5(y)} |\u03c8\u27e9)\u2020 = \u27e8\u03c8| {T \u2020\u03b3\u03f5(y), T \u2020\u00b5\u03bd(x)} |\u03c8\u27e9= \u27e8\u03c8| {T\u00b5\u03bd(x), T \u03b3\u03f5(y)} |\u03c8\u27e9 (3.16)Q =\u222bdxf\u00b5\u03bd(x)T\u00b5\u03bd(x) (3.17)|Q |\u03c8\u27e9|2 =\u222bdxdyf\u00b5\u03bd(x)f\u03b3\u03f5(y) \u27e8\u03c8|T\u00b5\u03bd(x)T \u03b3\u03f5(y) |\u03c8\u27e9 \u2265 0 (3.18)For arbitrary f\u00b5\u03bd(x), this demonstrates positive semi-definiteness for N ,though some care must be taken in picking a basis in which it is positive-definite when using its inverse. From this, one can write out the N\u00b5\u03bd\u03b3\u03f5(x, y)kernel as given by splitting the stress-energy tensor in terms of its expec-tation value \u27e8T\u00b5\u03bd(x)\u27e9 and a part with zero expectation value denoted byt\u00b5\u03bd(x) (10.11 in [6]):313.3. Stochastic Gravity from Schwinger-Keldysh Functional FormalismN\u00b5\u03bd\u03b3\u03f5(x, y) =12\u27e8{T\u00b5\u03bd(x), T \u03b3\u03f5(y)}\u27e9\u2212 \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T \u03b3\u03f5(y)\u27e9 = 12\u27e8{t\u00b5\u03bd(x), t\u03b3\u03f5(y)}\u27e9(3.19)This is the definition of the noise kernel. TheHS kernel is also completelyreal, though, unlike the noise kernel, the time ordering will matter due toantisymmetry.There are three more second derivative terms that can be found by sub-stituting in for h\u2212 in one or the other, or both, of the h+ in the previousexpression. One will find that the positive and negative action variationswill only be non-zero when taking the variation with respect to a metric per-turbation with the same sign. Furthermore, there is a symmetry betweenopposite terms so only a few modifications must be made to find the ex-pression with opposite signs. This means only one other expression needs tobe found, that being the second-derivative with respect to both the positiveand negative metrics:\u03b42\u0393\u03b4g+\u00b5\u03bd(x)\u03b4g\u2212\u03b3\u03f5(y)=\u03b4\u03b4h\u2212\u03b3\u03f5(y)[e\u2212i\u0393\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u00b5\u03bd(x))]= \u2212ie\u22122i\u0393[\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u00b5\u03bd(x))][\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S\u2212\u03d5\u03b4h\u2212\u03b3\u03f5(y))]+ ie\u2212i\u0393\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b4S+\u03d5\u03b4h+\u00b5\u03bd(x))(\u03b4S\u2212\u03d5\u03b4h\u2212\u03b3\u03f5(y))+ e\u2212i\u0393\u222bD\u03d5 eiS+\u03d5 \u2212iS\u2212\u03d5 ( \u03b42S+\u03d5\u03b4h+\u00b5\u03bd(x)\u03b4h\u2212\u03b3\u03f5(y))(3.20)The last term will go to zero as there is no part of the action that dependson both positive and negative metric perturbations. The remaining twosecond derivative expressions to be calculated are related to the ones thathave been shown in equations (3.7) and (3.20) with only a few minus signdifferences in certain places.The first term in (3.20) is very similar to the first term in (3.7) with theonly change being a negative sign in the second factor. The second termis now changed because there is no longer time ordering as in semiclassicalgravity. Instead, the orderings are as shown, simply based on the orderingof the derivatives:323.4. The Noise Kernel as a Stochastic Source4\u221a\u2212g(x)\u221a\u2212g(y) \u03b42\u0393\u03b4g+\u00b5\u03bd(x)\u03b4g\u2212\u03b3\u03f5(y)= \u2212i \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T \u03b3\u03f5(y)\u27e9+ i \u27e8T\u00b5\u03bd(x)T \u03b3\u03f5(y)\u27e9= \u2212i \u27e8T\u00b5\u03bd(x)\u27e9 \u27e8T \u03b3\u03f5(y)\u27e9+ 12\u27e8{T\u00b5\u03bd(x), T \u03b3\u03f5(y)}\u27e9+ i2\u27e8[T\u00b5\u03bd(x), T \u03b3\u03f5(y)]\u27e9= iN\u00b5\u03bd\u03b3\u03f5(x, y)\u2212H\u00b5\u03bd\u03b3\u03f5A (x, y)(3.21)H\u00b5\u03bd\u03b3\u03f5A (x, y) = \u2212i2\u27e8[T\u00b5\u03bd(x), T \u03b3\u03f5(y)]\u27e9 (3.22)At this point, one can consider the effective action by constructing anexpression that returns these derivatives. By taking the background metricto be the Minkowski metric and defining [h\u00b5\u03bd(x)] = h+\u00b5\u03bd(x) \u2212 h\u2212\u00b5\u03bd(x) and{h\u00b5\u03bd(x)} = h+\u00b5\u03bd(x) + h\u2212\u00b5\u03bd(x) one can write out \u0393 in the form (10.41 in [6]):\u0393 =12\u222bd4x \u27e8T\u00b5\u03bd(x)\u27e9 [h\u00b5\u03bd(x)]\u2212 18\u222bd4x d4y ([h\u00b5\u03bd(x)](H\u00b5\u03bd\u03b3\u03f5S (x, y) +H\u00b5\u03bd\u03b3\u03f5A (x, y) +K\u00b5\u03bd\u03b3\u03f5(x, y)){h\u03b3\u03f5(y)}\u2212 i[h\u00b5\u03bd(x)]N\u00b5\u03bd\u03b3\u03f5(x, y)[h\u03b3\u03f5(y)]) + ...(3.23)It is clear that in the limit h+ = h\u2212 = h, this expression completelyvanishes, though the derivatives still remain. For example, when takingthe first variation, only the \u27e8T\u00b5\u03bd(x)\u27e9 survives, which reproduces the semi-classical result. The [h\u03b3\u03f5(y)] factors act like classical sources as used inQFT. They vanish in the limit, but by taking their variations, the kernelscan be directly extracted. Finding two-point correlation functions will giveN\u00b5\u03bd\u03b3\u03f5(x, y), and certain other procedures may result in expressions in termsof the other kernels. This does not mean that the perturbations themselvesvanish, but they still must be small for the perturbation theory to be valid.3.4 The Noise Kernel as a Stochastic SourceIn the expression for the stochastic effective action, there is still an imaginaryterm containing the noise kernel. The noise kernel was defined in such a wayso that it is always real, so this term is purely imaginary. The identification333.4. The Noise Kernel as a Stochastic Sourceof this term as a stochastic source can be found from invoking an identityfrom Feynman and Vernon [32]. This relationship allows one to redefine thisterm such that it is real, and as a result forms the basis of stochastic gravity.The \u2019influence functional\u2019 is defined as ei\u0393, much in the same way thegenerating functional Z is defined:ei\u0393 = eiRe(\u0393)\u2212Im(\u0393) (3.24)e\u2212 Im(\u0393) = exp(\u221218\u222bd4x d4y [h\u00b5\u03bd(x)]N\u00b5\u03bd\u03b3\u03f5(x, y)[h\u03b3\u03f5(y)])(3.25)This can be written in terms of a functional integral over some variable\u03be\u00b5\u03bd (10.43 in [6]):\u222bD\u03beP [\u03be] exp(i2\u222bd4x\u221a\u2212g(x)\u03be\u00b5\u03bd(x)[h\u00b5\u03bd(x)])(3.26)The integral is written with some probability function P [\u03be] and a sourceterm that depends on [h\u00b5\u03bd(x)] as the source that vanishes in the limit. Theparticular form of the P [\u03be] function that recovers the corresponding term inthe effective action is given by:P [\u03be] = N exp(\u221212\u222bd4x d4y\u221a\u2212g(x)\u221a\u2212g(y)\u03be\u00b5\u03bd(x)N\u22121\u00b5\u03bd\u03b3\u03f5(x, y)\u03be\u03b3\u03f5(y))(3.27)Here, N is some normalization for the probability. Using a condensednotation where each index a and b respectively corresponds to integrationover x and y, and indices \u00b5\u03bd and \u03b3\u03f5, for example with arbitrary tensors:F afa =\u222bd4xF\u00b5\u03bd(x)f\u00b5\u03bd(x), Gabga =\u222bd4x d4y G\u00b5\u03bd\u03b3\u03f5(x, y)g\u03b3\u03f5(y)(3.28)One can show the Feynman-Vernon identity using a complete-the-squareproof in this notation:i2\u03bea[ha]\u2212 12\u03beaN\u22121ab \u03beb = \u221212(\u03bea + \u03bba)N\u22121ab (\u03beb + \u03bbb) +12\u03bbaN\u22121ab \u03bbb (3.29)i4\u03bea[ha] = \u221212\u03beaN\u22121ab \u03bbb,i4\u03beb[hb] = \u221212\u03bbaN\u22121ab \u03beb (3.30)343.4. The Noise Kernel as a Stochastic Source\u03bbb = \u2212 i2Nab[ha], \u03bba = \u2212 i2Nab[hb] (3.31)e\u2212 Im(\u0393) = exp(\u221218[ha]Nab[hb])\u222bD\u03be\u2032N exp(\u221212\u03be\u2032aN\u22121ab \u03be\u2032b)= exp(\u221218\u222bd4x d4y [h\u00b5\u03bd(x)]N\u00b5\u03bd\u03b3\u03f5(x, y)[h\u03b3\u03f5(y)]) (3.32)Here, I redefined \u03be with \u03be\u2032 = \u03be + \u03bb and defined the normalization con-stant N so that it cancels the determinant given by the Gaussian integral.This shows one can replace this factor in the influence functional with thefunctional integral over \u03be. The effective action can be rewritten with thisterm, though now one must average over the stochastic tensor to get theinfluence functional (10.46 in [6]):\u0393 = Re(\u0393) +12\u222bd4x\u221a\u2212g(x)\u03be\u00b5\u03bd(x)[h\u00b5\u03bd(x)] + ... (3.33)\u27e8ei\u0393\u27e9s = eiRe\u0393\u222bD\u03be P [\u03be] exp(12\u222bd4x\u221a\u2212g(x)\u03be\u00b5\u03bd(x)[h\u00b5\u03bd(x)])+ ...(3.34)Expectation values over the \u03be field are given a subscript s to indicatethey are a statistical average over the fluctuations. Given a Gaussian prob-ability function, the \u03be variable is a stochastic tensor sourced from the metricperturbations. The correlation functions can be found from the functionalintegral in the limit as [h\u00b5\u03bd(x)] goes to zero. For example, in a Minkowskibackground with\u221a\u2212g(x) = 1:\u27e8\u03be\u03b1\u03b2(x)\u27e9s =\u222bD\u03be exp(\u221212\u222bd4x d4y \u03be\u00b5\u03bd(x)N\u22121\u00b5\u03bd\u03b3\u03f5(x, y)\u03be\u03b3\u03f5(y))\u03be\u03b1\u03b2(x) = 0(3.35)\u27e8\u03be\u03b1\u03b2(z)\u03be\u03c3\u03c1(w)\u27e9s=\u222bD\u03be exp(\u221212\u222bd4x d4y \u03be\u00b5\u03bd(x)N\u22121\u00b5\u03bd\u03b3\u03f5(x, y)\u03be\u03b3\u03f5(y))\u03be\u03b1\u03b2(z)\u03be\u03c3\u03c1(w)= N\u03b1\u03b2\u03c3\u03c1(z, w)(3.36)353.5. Predictions of Stochastic GravityThe expectation value here is interpreted as being over all possible config-urations of the stochastic field \u03be. While this maintains the classical behaviorof the stress-energy tensor, the similarity to the quantum interpretation isapparent. Despite this, there is no \u2019quantization\u2019 of gravity here as the fieldis still treated as classical, not as a quantum field.These expressions give the relationship needed to understand stochas-tic gravity as it is clear that \u03be\u03b1\u03b2(x) corresponds with t\u03b1\u03b2(x) through thedefinition of the noise kernel:12\u27e8{t\u03b1\u03b2(x), t\u03c3\u03c1(y)}\u27e9s = N\u03b1\u03b2\u03c3\u03c1(x, y) = \u27e8\u03be\u03b1\u03b2(x)\u03be\u03c3\u03c1(y)\u27e9s (3.37)The addition of this stochastic part to the stress-energy tensor in thesemiclassical equation allows one to define a new equation that can describethe relationship between a stochastic stress-energy tensor and the gravita-tional field by expanding it out in the Einstein field equations. This is calledthe Einstein-Langevin Equation:G\u00b5\u03bd(x) = \u03ba( \u27e8T\u00b5\u03bd(x)\u27e9+ \u03be\u00b5\u03bd(x)) (3.38)3.5 Predictions of Stochastic GravityThe result of stochastic gravity is that one can introduce a stochastic com-ponent to the stress-energy tensor, and consequently relate this to curvedspacetime as described by the Einstein-Langevin equation. More generally,one can consider this concept as being a classical stochastic fluctuation tothe metric - meaning these fluctuations can be considered as physical andclassical, not just superpositions of uncertain quantum fluctuations. By therelationship between the stress-energy tensor and the Einstein tensor, onecan therefore infer that these fluctuations may also apply to the curvature.Much of the study of stochastic gravity begins with the Einstein-Langevinequation. This allows one to solve specific cases in general relativity follow-ing the same procedure as in classical or semiclassical gravity, then demon-strating how the result also depends on fluctuations. This method has beenused to study several different problems.Stochastic gravity has been used to study black holes, particularly inthe context of fluctuations due to quantum phenomena. Wu and Ford [35]looked at the fluctuations in radiation by studying the higher-order termsof the stress-energy tensor expectation value. Hu and Roura [36] consideredthis idea using stochastic gravity as the basis for study. In this work, they363.5. Predictions of Stochastic Gravityused the Einstein-Langevin equation to demonstrate the dependence of theblack hole evaporation on the stochastic component.Astrophysical phenomena, particularly relativistic stars, also benefit fromthe tools of stochastic gravity. Satin [37] devised a formalism for applyingthe Einstein-Langevin equation to the study of relativistic stars. Startingfrom the semiclassical starting point of assuming a perfect fluid with someequation of state, the stochastic component can then be separated out andrelated to a stochastic component of the curvature. This work may be usefulin studying the fluctuations in fluid velocity, generalization to more complexsituations one may find in relativistic stars such as rotation, and ultimatelya better understanding of the stability of these objects.Cosmology is one of the primary areas of study where understandingthe effect of quantum phenomena on spacetime is vitally important. Thebehavior of the quantum fluctuations as described by the noise kernel canbe studied in particular cosmological models such as de Sitter spacetime [6].Stochastic gravity can be used to calculate the effects of one-loop correctionsto calculations of cosmological perturbation as a result of inflation [64].3.5.1 Dark EnergyOne of the primary motivations in this thesis will be studying the stochasticgravity explanation of \u2019dark energy\u2019. Wang et al. [25] demonstrated that byassuming a metric with the form of the FRW metric with a spatial depen-dence as well as the time dependence, the Einstein-Langevin equation canbe used to derive a parametric resonance that can be tuned to give resultsconsistent with the presence of \u2019dark energy\u2019.It can be shown that by imposing a high-energy cutoff and allowing thebreaking of Lorentz-invariance, the stress-energy tensor takes the form givenin (2.29). The fluctuation in the energy T00 is given by:\u27e8(T00 \u2212 \u27e8T00\u27e9)2\u27e9 = 23\u27e8T00\u27e92 (3.39)Along with this, the variation of the expectation value of the squareof the energy density can be plotted as a function of time and distance inunits of the cutoff. This illustrates the inhomogeneity of the vacuum, anddemonstrates how correlations exist on the scale of the cutoff, then becomingmore diluted over larger times and distances. By assuming the metric takesthe form:ds2 = \u2212dt2 + a2(t, x\u20d7)(dx2 + dy2 + dz2) (3.40)373.5. Predictions of Stochastic GravityThe Einstein field equations can be solved precisely for relations betweena(t, x\u20d7) and the stress-energy tensor. In particular, it can be shown that:a\u00a8+\u21262(t, x\u20d7)a = 0, \u21262(t, x\u20d7) = \u03baT00 +\u03baa2(Txx + Tyy + Tzz) (3.41)This is the form of a parametric oscillator where \u21262 is a quasiperiodicfunction that will be dependent on the fluctuations in the stress-energy ten-sor. Solutions for this are found to give a parametric resonance effect. Thismeans that as spacetime fluctuates, there is a slight trend toward expansion,meaning an accelerated expansion rate can be calculated based on the pa-rameters of the model. As the stress-energy tensor with this regularizationmethod is dependent on the cutoff, this expansion rate will be dependent onthe cutoff. Furthermore, each quantum field will contribute to this process,meaning that if one tunes the cutoff and the number of fields, a particularvalue of the apparent Hubble expansion can be extracted.Initially, this appears to suffer a similar problem to the fine-tuning prob-lem of the standard semiclassical approach, but in fact, the parameters aremuch more physically reasonable in this case. In the semiclassical case, thecutoff had to be reduced to unphysical levels to explain the apparent cos-mological constant. Wang et al. [25] suggest a form H \u223c \u039be\u2212\u03b2\u221aG\u039b for theHubble expansion rate. According to their initial calculations, a cutoff about1000 times the Planck scale and two scalar fields would give the observedvalue of the apparent Hubble expansion rate.Cree at al. [38] revised some of the analysis primarily through numer-ical methods. The accuracy of the model for a low number of fields wasquestioned, though a higher number of fields were demonstrated to be morevalid. In particular, they estimated that if the cutoff is assumed to be atthe Planck energy, 6000 fields are needed. A slightly higher value for thecutoff allows the number of fields to be reduced. Overall, these parametersare still in a physically reasonable range, and not as heavily fine-tuned as acosmological constant.3.5.2 QuestionsIn this chapter, I discussed stochastic gravity and some of the problemsthat it can be used to look at. The most notable of these was the predictionthat a parametric resonance between quantum fields and the fluctuatingcurvature could allow for an accelerating expansion of the vacuum, whichcould correspond to Dark Energy. For the rest of this thesis, I want to focus383.5. Predictions of Stochastic Gravityon a number of problems that could arise from stochastic gravity and howI would like to study these problems. The two main issues I would like toaddress are:\u2022 Geodesic Scattering - Do the fluctuations in the metric cause scat-tering? The geodesics through a fluctuating spacetime may shift as thelengths of paths change, causing a slight broadening and a scatteringeffect. If these scattering effects are significant enough, can they addup to a large amount over long distances? How do the physical fieldsin that spacetime react to the scattering? Is there some dependenceon their wavelength? Are there methods based on studying geodesicsdirectly that can give similar results to those of Wang et al. [25]? Thiswill be the focus of Chapter 4.\u2022 Physically Regulated Fields - Can one use Pauli-Villars regular-ization to physically regulate a quantum field theory in curved space-time? In the work of Wang et al. [25], high-energy cutoff regularizationwas used, which breaks Lorentz Invariance. Can a manifestly Lorentz-Invariant regularization approach such as Pauli-Villars be used to solvethe same problem and give physically reasonable results? This will bethe focus of Chapter 5.39Chapter 4Geodesic Scattering4.1 Justification for ScatteringIn stochastic gravity, the concept of fluctuating spacetime is introduced.Fluctuations can lead to scattering, which may be an important predictionof stochastic gravity if the predicted scattering becomes much larger thanobserved. To understand why one may expect scattering in this case, I willoutline some physical justification from other effects in physics.4.1.1 Multiple ScatteringIn the study of nuclear physics and particle physics, the use of scattering ex-periments has been an important tool for understanding the composition ofmatter. Nuclear physics experiments involve the scattering of nuclei throughtargets and the understanding of this effect is vital to interpret the results[65, 66]. Much of the study of these experiments involves quantifying theloss of energy to these materials and scattering by the many scatterers inthe path of the nuclei. The result of scattering off of many scatterers in thissort of experiment is called multiple scattering [66].Multiple scattering has been studied extensively and is a major partof determining the broadening of signals in such experiments. The primaryeffect that is considered to contribute is the elastic scattering of nuclei due toRutherford Scattering. This effect can be calculated using only the Coulombforce between a nucleus with charge Z1 and an incoming charged particlewith charge Z2. The differential cross-section of scattering is given by:d\u03c3d\u2126=Z1Z2\u03b1\u210fc4Ekcsc4(\u03b82) (4.1)The first step in understanding this process is getting a single-scatteringcross-section. In the case of Rutherford scattering, that is described by(4.1). Once single-scattering is understood, multiple-scattering can be usedto understand the behavior over long distances with many scattering events.Many simplified properties appear in multiple scattering, such as a tendency404.1. Justification for ScatteringFigure 4.1: An example diagram of many small single-scattering events lead-ing to a wide normal distribution in scattered angletowards a Gaussian distribution. At this point, scattering grows with thesquare root of distance, similar to a random walk. While the expectationvalue remains at zero scattering, the width of the distribution (described bythe standard deviation) grows, such as in figure 4.1.The important concept here is that a small effect can be described by amuch simpler statistical relationship when many events are combined, suchas over a long distance. Given that stochastic gravity predicts that thereare extremely weak Planck-scale effects, it may be that these effects becomenoticeable on cosmological scales. I will first look at the effects of waves andscattering by considering Rayleigh scattering and single scattering.4.1.2 Rayleigh ScatteringRutherford scattering describes the scattering of a charged particle wherethe influence of coulomb forces from individual scatterers is considered. Thesame may be true of Rayleigh scattering; individual scatterers in a mediummay cause the scattering of light. One can alternatively consider the scat-tering to be due to variations in the dielectric constant due to the scatterers[67]. This view gives a much closer problem to the one given by a fluctuatingspacetime.As a starting point, one can consider Mie theory which describes thescattering of light due to spheres. A Born approximation can be used tofind an approximate expression for the scattering amplitude for an incomingplane wave with incoming wave vector k\u20d7 = kn\u02c6i scattering off a sphericallysymmetric potential \u03b3 with finite size R to give an outgoing wave vectork\u20d7 = kn\u02c6s (pg. 173-176 in [68]):A(n\u02c6s, n\u02c6i) =\u222bd3x exp(ik(n\u02c6s \u2212 n\u02c6i) \u00b7 x\u20d7)\u03b3(x\u20d7) (4.2)414.2. Single Scattering with GravitonsA(\u03b8) =2\u03c0k\u03b3sin(\u03b82) \u222b R0sin(2kr sin(\u03b82))rdr \u223c k2\u03b3V (kR << 1) (4.3)The scattering probability |A|2 is then dependent on k4 and thus propor-tional to \u03bb\u22124. This approach appears to assume a molecular composition,however, one can consider that this is just as valid for continuous fluctua-tions. As long as the length scales of the fluctuations are much smaller thanthe wavelength, the kR << 1 approximation applies, and the \u03bb\u22124 result isvalid. If one Fourier transforms \u03b3(x\u20d7), it is clear that the scattering ampli-tude is directly related to \u03b3\u02dc(k(n\u02c6s \u2212 n\u02c6i)). Rayleigh scattering would then berelated to fluctuations given by a power spectrum \u03b3\u02dc(k) \u223c k2.From these examples, a few questions emerge about the scattering presentin stochastic gravity. If fluctuations cause scattering, how can one calculatescattering amplitudes due to these fluctuations? Multiple scattering sug-gests that the exact form may not matter, instead, if one has many smallscattering events that add up over a large distance, one can approximateto a Gaussian distribution which scales predictably with distance. Still, onemust find the magnitude of the scattering amplitude to be able to make theseapproximations. To get these approximations, I will first consider methodsfor calculating the scattering of waves.4.2 Single Scattering with GravitonsThe first approach I will demonstrate is an approach to calculate a single-scattering cross-section with the \u2019In-Out\u2019 formalism of QFT. To see if singlescattering of waves can occur from vacuum fluctuations using a perturbativeapproach, one can look at the scattering for a particle in a field interactingwith vacuum fluctuations. This is the sort of approach one would take inother interacting theories such as Quantum Electrodynamics to calculatescattering amplitudes. The way to do this in the formalism introduced inChapter 2 is to calculate the correlation function:\u27e8\u2126| T {\u03d5(x)\u03d5(y)} |\u2126\u27e9 =\u222bD\u03d5Dh eiSEH+iS\u03d5\u03d5(x)\u03d5(y)=\u222bD\u03d5Dh eiS(0)EH+iS(0)\u03d5 eiS(1)\u03d5 eiS(2)\u03d5 ... \u03d5(x)\u03d5(y)(4.4)In this formalism, one can expand out the interaction terms and findexpectation values in the free theory in Minkowski spacetime. In this case,424.2. Single Scattering with Gravitonsthe waves are described by the scalar field. Including a functional integralover the gravitational perturbation, one may use the graviton propagator.Given that there are serious doubts about the validity of any sort of quantumgravity, this should not be taken too seriously except to demonstrate thefollowing point.Using these tools, the Feynman rules are such that S(1)\u03d5 will give a vertexwith two \u03d5 and a graviton operator h\u00b5\u03bd , while S(2)\u03d5 will give a vertex with two\u03d5 and two gravitons. There are also self-interactions between the gravitons,bringing a higher order in the diagram. Including a factor of \u03b2 for eachfactor of h\u00b5\u03bd allows one to indicate the order by counting the number ofvertices. Following these rules, the diagrams that can be given up to \u03b24 aregiven in figure 4.2.As an example, figure 4.2 b) would be one of the diagrams coming from:\u222bD\u03d5Dh eiS(0)EH+iS(0)\u03d5 (\u221212(S(1)\u03d5 )2)\u03d5(x)\u03d5(y) (4.5)At the next order, figure 4.2 d) would be one of the diagrams comingfrom: \u222bD\u03d5Dh eiS(0)EH+iS(0)\u03d5 124(S(1)\u03d5 )4 \u03d5(x)\u03d5(y) (4.6)Note that I have only included one example diagram per term in theexpansion; a full use of Wick\u2019s theorem and other calculation methods wouldgive a broader set of results. In each case, the action for \u03d5 is given in equation(2.25), and by integrating over both \u03d5 and h, one gets propagators for thequantum field and graviton propagators, as shown in the diagrams.In each of these diagrams, momentum is conserved at each vertex. Thismeans that no momentum will be transferred to the \u03d5 traveling from xto y. This means that none of these processes can lead to any scattering,at least not in the usual interpretation of quantum field theory. The lackof contribution from such vacuum diagrams is a standard result in otherquantum field theories in flat spacetime, such as QED. While this alreadydemonstrates a significant issue with this approach, I will also look at how,if at all, stochastic gravity could change this result.4.2.1 The Stochastic Gravity ApproachThe effect of metric fluctuations is derived from stochastic gravity, and itfollows that as a result, the formalism of stochastic gravity must be used to434.2. Single Scattering with Gravitons(a) O(\u03b20) (b) O(\u03b22)(c) O(\u03b24) (d) O(\u03b24)(e) O(\u03b24) (f) O(\u03b24)(g) O(\u03b24) (h) O(\u03b24)Figure 4.2: Feynman diagrams corresponding to \u27e8\u2126| T {\u03d5(x)\u03d5(y)} |\u2126\u27e9 ex-panded up to O(\u03b24).444.2. Single Scattering with Gravitonssolve problems like this. I will look at the case of an extra field in a space-time with fluctuations described by a noise kernel which has already beendetermined. The effective action, in this case, will be given by combiningthe gravitational action, the scalar action, and the effective potential:Seff [g\u00b1, \u03d5\u00b1] = Sg[g+]\u2212 Sg[g\u2212] + S\u03d5[\u03d5+, g+]\u2212 S\u03d5[\u03d5\u2212, g\u2212] + \u0393[g\u00b1] (4.7)In this case, one can assume that the field integrated out to find \u0393 isdifferent from the one that is being scattered given as \u03d5. There are severalterms in the effective potential \u0393 given in (3.23), though in this case I willonly focus on the noise kernel term, which is given by:\u0393N [h\u00b1] =i8\u222bd4x d4y [hab(x)]Nabcd(x, y)[hcd(y)] (4.8)The effective action can be expanded out to calculate the scattering am-plitude. I will calculate one particular diagram for the two-point correlationfunction of \u03d5 - that being similar to figure 4.2 d. The noise kernel corre-sponds to a loop between two points connected to graviton vertices, muchlike in that diagram. I will assume that \u03d5 is massless in order to simplifythe calculations:\u222bD\u03d5+D\u03d5\u2212 eiS(0)\u03d5 [\u03d5+]\u2212iS(0)\u03d5 [\u03d5\u2212](i\u0393[h\u00b1])12(iS(1)\u03d5 [\u03d5+]\u2212 iS(1)\u03d5 [\u03d5\u2212])2\u03d5+(x1)\u03d5+(x2)=116\u222bd4y1 d4y2 d4z1 d4z2 (h+ab(z1)\u2212 h\u2212ab(z1))Nabcd(z1, z2)(h+cd(z2)\u2212 h\u2212cd(z2))F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1( \u27e8\u2202\u00b5\u03d5+(y1)\u2202\u03bd\u03d5+(y1)\u2202\u03b3\u03d5+(y2)\u2202\u03f5\u03d5+(y2)\u03d5+(x1)\u03d5+(x2)\u27e9h+\u03b1\u03b2(y1)h+\u03c3\u03c1(y2)\u2212 \u27e8\u2202\u00b5\u03d5\u2212(y1)\u2202\u03bd\u03d5\u2212(y1)\u2202\u03b3\u03d5+(y2)\u2202\u03f5\u03d5+(y2)\u03d5+(x1)\u03d5+(x2)\u27e9h\u2212\u03b1\u03b2(y1)h+\u03c3\u03c1(y2)\u2212 \u27e8\u2202\u00b5\u03d5+(y1)\u2202\u03bd\u03d5+(y1)\u2202\u03b3\u03d5\u2212(y2)\u2202\u03f5\u03d5\u2212(y2)\u03d5+(x1)\u03d5+(x2)\u27e9h+\u03b1\u03b2(y1)h\u2212\u03c3\u03c1(y2)+ \u27e8\u2202\u00b5\u03d5\u2212(y1)\u2202\u03bd\u03d5\u2212(y1)\u2202\u03b3\u03d5\u2212(y2)\u2202\u03f5\u03d5\u2212(y2)\u03d5+(x1)\u03d5+(x2)\u27e9)h\u2212\u03b1\u03b2(y1)h\u2212\u03c3\u03c1(y2))(4.9)The principles for simplifying this expression are piecewise not too com-plicated. There will be correlation functions between positive and negativemetric perturbation fields, but not between the positive and negative \u03d5 fieldsin the free theory. It is also necessary to remove disconnected diagrams. Asthere are no correlators between positive and negative field operators andeach vertex must either have one or the other, there is no way to make a454.2. Single Scattering with Gravitonsconnected diagram except for with the first correlation function where thereare only \u03d5+ contributions.The next step will be to calculate metric correlations for the remainingterm. I will assume that there is no self-interaction in this case and onlycalculate two-point correlation functions. In order for the diagrams to beconnected, the y points must connect to the z points and not each other.In this case, the correlators between the metric perturbations will act tomandate time-ordering for the metric propagators so that the sourcing bythe noise kernel always comes before the scattering with the \u03d5 field, whichmakes physical sense as it is interpreted as the source of the scattering. Asan example, where there is a propagation between y1 and z1 and betweeny2 and z2 (See [6] eq. 9.63 for metric perturbation correlation functions):\u27e8in| P{(h+ab(z1)\u2212 h\u2212ab(z1))h+\u03b1\u03b2(y1)} |in\u27e9 \u27e8in| P{(h+cd(z2)\u2212 h\u2212cd(z2))h+\u03c3\u03c1(y2)} |in\u27e9= \u27e8in| P{h+ab(z1)h+\u03b1\u03b2(y1)\u2212 h\u2212ab(z1)h+\u03b1\u03b2(y1)} |in\u27e9\u27e8in| P{h+cd(z2)h+\u03b1\u03b2(y1)\u2212 h\u2212ab(z1)h+\u03b1\u03b2(y1)} |in\u27e9= (\u27e8in| T {hab(z1)h\u03b1\u03b2(y1)} |in\u27e9 \u2212 \u27e8in|hab(z1)h\u03b1\u03b2(y1) |in\u27e9)(\u27e8in| T {hcd(z2)h\u03c3\u03c1(y2)} |in\u27e9 \u2212 \u27e8in|hcd(z2)h\u03c3\u03c1(y2) |in\u27e9)= \u03b8(y01 \u2212 z01)\u03b8(y02 \u2212 z02) \u27e8in| [h\u03b1\u03b2(y1), hab(z1)] |in\u27e9 \u27e8in| [h\u03c3\u03c1(y2), hcd(z2)] |in\u27e9(4.10)The form given here is just the retarded propagator for the gravitonsso that the stochastic tensor is sourced before the scalar interaction vertex.While the exact form of these propagators is not immediately clear, it givessome limits that were not present in the previous attempt at this problem.In particular, the Feynman propagator will allow sourcing of the gravitonsfrom all points in spacetime, while the retarded propagator makes more sensein studying a system evolving in time. With this result, the propagatorsgenerally take the form:\u03b8(y01 \u2212 z01)\u03b8(y02 \u2212 z02)\u222bd4p(2\u03c0)4(e\u2212ip(y1\u2212z1) \u2212 eip(y1\u2212z1))\u00d7\u222bd4q(2\u03c0)4(e\u2212iq(y2\u2212z2) \u2212 eiq(y2\u2212z2))(4.11)As a next step, one can consider the two-point correlation function ofthe stress-energy tensor perturbations \u03be\u00b5\u03bd(x). The noise kernel is includedin the expansion, but as shown by the definition in Chapter 3, this can be464.2. Single Scattering with Gravitonsreplaced by an expectation value of two stochastic tensors at each of thespacetime points:N\u00b5\u03bd\u03b1\u03b2(x, y) =\u222bD\u03beP [\u03be]\u03be\u00b5\u03bd(x)\u03be\u03b1\u03b2(y) (4.12)As the Fourier transform is used throughout this calculation, it is usefulto find the Fourier transforms of the stochastic tensors. The Fourier trans-forms can then be taken in the expectation value and the result is as shownby (See [6] eq. 11.42):\u27e8\u03be\u02dc\u00b5\u03bd(p)\u03be\u02dc\u03b1\u03b2(q)\u27e9 = (2\u03c0)4\u03b4(4)(p+ q)N\u00b5\u03bd\u03b1\u03b2(p) (4.13)This demonstrates that the fluctuations will still maintain momentumconservation, even if there are fluctuations in the processes described byN\u00b5\u03bd\u03b1\u03b2(p). Even in elastic scattering, some momentum transfer is needed -meaning this approach will not give scattering. This makes sense in thatthere is nothing new in the calculations compared to semiclassical gravity inthis regard - this is essentially the same diagram as figure 4.2 d. The onlynew addition is that the vacuum bubble vertices must be at an earlier timethan the \u03d5 vertices they connect to, though the conservation of momentumis unchanged.4.2.2 Interpretation of Wave ScatteringIn the previous section, it has been made clear that the QFT approach to thescattering of waves in a fluctuating spacetime will not work. While there areslight changes in the calculations between semiclassical gravity, which followsthe standard conventions of QFT more closely, and stochastic gravity, wherethe Schwinger-Keldysh formalism explicitly introduces retarded propagatorsfor the gravitons, the important aspects of these calculations are the samein both. It is a prediction of QFT that waves propagating in spacetime arenot affected by vacuum terms.Fluctuations introduce a breaking of translation symmetry, which inprinciple should allow the breaking of the conservation of momentum onsmall scales. This is an example of the uncertainty principle. While thesefluctuations should average to zero, as the lack of scattering suggests, ifthey can have cumulative effects in a random walk, this average may notbe a good descriptor. If these fluctuations are assumed to be \u2019real\u2019 in someway, as stochastic gravity suggests, the averaging out may not give the fullpicture. Ultimately, the necessary step to calculating a multiple scattering474.3. Geodesic Deviationdistribution is a single scattering amplitude, which does not appear possibleusing the calculation methods shown here.The QFT description includes gravitons as particles in the traditionalsense. In GR, however, the curvature affects geodesics directly with noapparent particle nature. The behavior of the waves is dictated by thegeodesics which are known to be affected by these fluctuations. It may bethat the use of gravitons in QFT requires a different approach to calculationsthan what is shown here.4.3 Geodesic DeviationWhile the standard quantum field theory approach to scattering has shownlittle promise, there has been work done on a different approach by focus-ing solely on the deviation of the geodesics themselves. Parikh et al. [69]demonstrated an approach to this, focusing on demonstrating the effect ofquantum graviton fluctuations on an interferometer detector like LIGO. Thiswas followed up by Cho and Hu [70] who studied the same system in a moregeneral way and formulate an approach using stochastic gravity formalism.I will explain how this problem was studied and suggest some approachesto geodesic scattering that may arise from this approach.4.3.1 Experimental Measurement of Geodesic ForceI will outline the approach used by Cho & Hu [70] to study the experimentalpossibility for the measurement of graviton noise, though this was done firstby Parikh et al. [69]. In their derivation, the perturbation was defined byg\u00b5\u03bd = \u03b7\u00b5\u03bd + \u03bah\u00b5\u03bd , with \u03ba2 = 16\u03c0G, so I will use their convention in thissection.The Einstein-Hilbert action in the transverse-traceless gauge is used asa basis for describing the gravitons. The corresponding Einstein-Hilbertaction is given by (eq. 4 in [70]):SEH =1\u03ba2\u222bd4x\u221a\u2212g(x)R(x) = \u221214\u222bd4x \u2202\u03b1h\u00b5\u03bd(x)\u2202\u03b1h\u00b5\u03bd(x) (4.14)To write out the perturbations in a useful form, a fact from Ford andParker [46] is used, where the perturbations can be written as the sum oftwo scalar fields corresponding to each of two polarizations. Writing out theperturbation in terms of these scalar fields h(s) (eq. 5 in [70]):484.3. Geodesic Deviationh\u00b5\u03bd(x) =\u222bd3p\u2211sh(s)(p\u20d7, t)\u03f5(s)\u00b5\u03bd (p\u20d7)eip\u20d7\u00b7x\u20d7 (4.15)By substituting this into the Einstein-Hilbert action and following somesteps to eliminate the dependence on the polarization, an action for theindividual scalar components is found (eq. 7 in [70]):SEH = \u221212\u222bd4x\u2211s\u2202\u03b1h(s)(x)\u2202\u03b1h(s)(x) (4.16)The interferometer system is set up to have two masses following geodesicswith worldlines Xi(t) and Y i(t). Fermi Normal Coordinates (z\u20d7, t) are usedto write the action in terms of the geodesic distance between the two world-lines, with Xi(t) being defined as stationary. The combination of these co-ordinates with the transverse-traceless gauge is approximately compatible,though a more complete description may require more precision in definingcoordinates and gauge in a consistent way. In these coordinates, one of themasses stays fixed, so that the action of the detector is just the position ofthe other mass given by:Sdet = \u2212m\u222b \u221a\u2212ds2 = \u2212m\u222b \u221a\u2212g\u00b5\u03bddY \u00b5dY \u03bd (4.17)Expanding the metric around the worldline Xi(t) to second order in z\u20d7in these coordinates (eq. 9-11 in [70]):g00(t, z\u20d7) = \u22121\u2212Ri0j0(t, 0)zi(t)zj(t) + ...g0i(t, z\u20d7) = \u221223R0ijk(t, 0)zj(t)zk(t) + ...gij(t, z\u20d7) = \u03b4ij \u2212 13Rikjl(t, 0)zk(t)zl(t) + ...(4.18)Substituting this into the detector action, where dY 0 = dt and dY i =dzi, with the Riemann tensor component Ri0j0(t, 0) = \u2212\u03ba2 h\u00a8ij (eq. 14 in[70]):ds2 = (\u22121\u2212Ri0j0(t, 0)zi(t)zj(t)) dt2 + \u03b4ijdzidzj + ...Sdet = \u2212m\u222b \u221adt2 +Ri0j0(t, 0)zi(t)zj(t)dt2 \u2212 \u03b4ijdzidzj + ...= \u2212m\u222bdt\u221a1 +Ri0j0(t, 0)zi(t)zj(t)\u2212 \u03b4ij z\u02d9iz\u02d9j + ...(4.19)494.3. Geodesic DeviationSdet =\u222bdt [m2\u03b4ij z\u02d9iz\u02d9j +m\u03ba4h\u00a8ij(t)zi(t)zj(t)] + ... (4.20)A few assumptions were made in both papers to get to this point. Infinding the Lagrangian in terms of the Fermi Normal Coordinates, the lowestorder terms in the metric were included, which are quadratic in z. If largerdistances are assumed, higher-order terms may come into consideration,which would be needed in a more complete description. This assumptionmeans that the wavelengths of the gravitons considered must be larger thanthe distance between the two masses.In the approach taken by Parikh et al. [69], one of the two polarizationsis chosen so as to simplify and demonstrate the calculation for one of the twofields. Cho and Hu [70] do not do this so as to keep the calculation general.As these fields are decoupled, it does not matter much for the analysis. Thegauge of the field will determine the exact form of the polarization tensor\u03f5\u00b5\u03bd , and in particular, the form in the transverse-traceless gauge is simplifieddue to the traceless condition.At this point, the approaches differ. Following the stochastic gravityderivation, Cho and Hu follow through with the approach described in Chap-ter 3 to find an influence action. The second term in the detector action isrewritten in terms of a source acting on the metric perturbation. The metricperturbations are integrated out, leaving an effective action on the geodesicdistance (eq. 25 in [70]):\u0393 =\u222bdtdt\u2032 [12(zi+(t)zj+(t)\u2212 zi\u2212(t)zj\u2212(t))Dijkl(t, t\u2032)(zk+(t\u2032)zl+(t\u2032) + zk\u2212(t\u2032)zl\u2212(t\u2032))+ i(zi+(t)zj+(t)\u2212 zi\u2212(t)zj\u2212(t))Nijkl(t, t\u2032)(zk+(t\u2032)zl+(t\u2032)\u2212 zk\u2212(t\u2032)zl\u2212(t\u2032))](4.21)This is reminiscent of (3.23), though in this context, the worldline ap-proach gives integrals over time. The noise kernel is interpreted in a similarway as in that case, and still has the relationship with the stochastic tensorsas given by (3.36), which can be found by applying the Feynman-Vernonidentity [32]. Using the notation [zi(t)zj(t)] = zi+(t)zj+(t) \u2212 zi\u2212(t)zj\u2212(t) and{zi(t)zj(t)} = zi+(t)zj+(t) + zi\u2212(t)zj\u2212(t):\u2329ei\u0393\u232as= exp{i2\u222bdtdt\u2032 [zi(t)zj(t)]Dijkl(t, t\u2032){zi(t\u2032)zj(t\u2032)}}\u222bD\u03beP [\u03be] exp{\u2212i\u222bdt \u03beij(t)[zi(t)zj(t)]} (4.22)504.3. Geodesic DeviationWhen finding the effective potential for the perturbations there wereterms with the difference between positive and negative perturbations. Inthis case, in the effective action for the geodesics, there are similar termsthat go to zero when the positive and negative geodesics are taken to be thesame.The effective action can be written by taking the stochastic effectiveaction to not be averaged over in \u03be - that is, use ei\u0393 and not\u2329ei\u0393\u232asasintegrated over in (4.22). The equation of motion for the geodesic can befound by taking this stochastic effective action with the rest of the non-interacting geodesics action and varying with respect to z+ and taking thelimit as z+ = z\u2212 = z (eq. 34, 35 in [70]):Seff [z+, z\u2212] =\u222bdtm2\u03b4ij(z\u02d9i+z\u02d9j+\u2212z\u02d9i\u2212z\u02d9j\u2212)+\u0393[z+, z\u2212],Seff [z+, z\u2212]zi+(t)\u2223\u2223\u2223\u2223z+=z\u2212=z= 0(4.23)mz\u00a8i(t) = \u22122\u03b4im\u222bdt\u2032Dmnkl(t, t\u2032)zn(t)zk(t\u2032)zl(t\u2032) + 2\u03b4ik\u03bekl(t)zl(t) (4.24)The first term on the right is a dissipation term, and this is small underthe assumptions of small z. The second term is the fluctuation term, relatedto the stochastic tensor \u03bekl(t). In this case, the expression is written in termsof time-dependent coordinates which describe the spatial coordinates. Choand Hu use this expression to solve the equation assuming an initial zl0(0) =(0, 0, z0) and z\u02d9l(0) = 0 in order to set boundary conditions. Correlationfunctions of the coordinate are then found by finding expectation valueswith the \u03be integral. First-order terms in \u03be will go away, though in second-order with \u03be, there will be a relationship with the noise kernel.4.3.2 Intrinsic and Induced Geodesic DeviationIn the approaches by Parikh et al. [69] and Cho & Hu [70], some assumptionsare made in the analysis that relates to the particular system that they arestudying. The two masses correspond to the ends of an interferometer suchas LIGO, which has light bouncing between the ends. As a result, thefocus is on this sort of experiment and the detectability of gravitons. It hasbeen argued by Dyson [71] that individual gravitons are not detectable, andthese results agree with this assessment. However, in trying to understandthese quantum effects on cosmic scales, the small-scale assumptions of theseresults may not be valid.514.4. Instability in Geodesic ScatteringIn both papers, a hard frequency cutoff is assumed based on the ex-perimental setup. In this case, this is valid because the cutoff is defined inrespect to a particular physical system which is well-defined, in particular interms of the wavelengths the system is sensitive to. If instead one shifts tothinking of this system in terms of physical effects in free space that are notindividually detected, a physical cutoff must be selected independent of theexperiment. In the case of Cho and Hu [70], the scalar graviton fields h(s) istreated as the source of the noise kernel, and the calculation is carried outfollowing this principle. These are known as \u2019intrinsic\u2019 metric fluctuations[6].In the work of Wang et al. [25], the source of the noise is assumed to bescalar quantum fields that cause fluctuations in the metric. The particularform of this noise is the one described in section 3.5.1. In this approach, thefields generating the fluctuations are taken into account in the calculationthrough the effective potential, which in turn can be used to calculate thefluctuations in the geodesics. These are known as \u2019induced\u2019 metric fluctua-tions [6].Both intrinsic and induced metric fluctuations in stochastic gravity havebeen studied [6], and so an approach that included one of the other, or both,would be valid. In combining the approaches of Parikh et al. [69] and Cho& Hu [70] with the assumptions of Wang et al. [25], one would insteadassume induced metric fluctuations as a starting point. Given a high cutoff,the induced fluctuations may be much stronger than the intrinsic fluctua-tions, which means there is an improved chance of detectability, though it islikely still very small. This may offer an alternative approach to calculatinggeodesic scattering from this process.4.4 Instability in Geodesic ScatteringWang et al. [25] demonstrated the possible effects of parametric oscillatorsin stochastic gravity. The approach taken by Parikh et al. [69] also containsthe possibility of parametric oscillation. The system they study reduces tothe classical action (eq. 25 in [69]):S =\u222bdt [12mh(h\u02d92 \u2212 \u03c92h2) + 12m0z\u02d92 \u2212 gh\u02d9z\u02d9z] (4.25)Here, h is the scalar component of a chosen polarization,mh is a constantrelated to the gravitational field, m0 is the particle mass of one of the endsof the detector (assumed to be small) and z is the geodesic separation.524.4. Instability in Geodesic ScatteringThe equations of motion can be derived in a straightforward way using theEuler-Lagrange equation:ddt\u2202L\u2202z\u02d9\u2212 \u2202L\u2202z= 0\u2192 z\u00a8 \u2212 gm0h\u00a8z = 0 (4.26)ddt\u2202L\u2202h\u02d9\u2212 \u2202L\u2202h= 0\u2192 h\u00a8+ \u03c92h\u2212 gmh(z\u00a8z + z\u02d92) = 0 (4.27)One can solve this perturbatively by assuming a small z, or a large mh:h\u00a80 + \u03c92h0 = 0, z\u00a8 \u2212 gm0h\u00a80z = 0 (4.28)The value of m0 is dependent on the mass of the classical particle thegeodesic corresponds to, while mh corresponds to the value of \u03ba and the sizeof the space assuming side lengths L [69]. Significant simplifications can bemade by taking limits apparent from the definitions of mh and g:mh =L316\u03c0\u210fG2\u2192\u221e, g = m02\u221a\u210fG\u2192 gm0=12\u221a\u210fG(4.29)If one assumes that the space is arbitrarily large, mh blows up, and sothe backreaction of z on h goes away. This is a valid limit in this caseas, while the system is defined in terms of a massive detector, geodesicsin free space should have no backreaction on the gravitational field. Thecoupling also does not depend on the mass of the interferometer, which isthe equivalence principle at work. This makes the perturbative relationshipexact. This is the form of a parametric oscillator where the solution to h isexactly sinusoidal:h(t) = Ah cos(\u03c9t+ \u03d50), z\u00a8 \u2212 Ah2\u221a\u210fGcos(\u03c9t+ \u03d50)z = 0 (4.30)Ignoring boundary conditions to put arbitrary amplitude and phase, thisis a particular form of parametric oscillator which can be described by theMathieu equation [72]:d2zdt2+ (\u03b4 + \u03f5cos(t))z = 0 (4.31)The stability of \u03b4 = 0 for some value of \u03f5 is of interest here. z is notbounded, so there is either instability where it grows exponentially or a534.4. Instability in Geodesic Scattering\u2019neutral stability\u2019 where it remains quasi-periodic. If the conditions for in-stability are met, Floquet theory gives a general solution by assuming somesolutions z1(t) and z2(t) are chosen to scale up after each period:z1(t+ T ) = \u00b51z1(t), z2(t+ T ) = \u00b52z2(t) (4.32)z1(t) = \u00b5tT1 P1(t), z2(t) = \u00b5tT2 P2(t) (4.33)Here, P1(t) and P2(t) are periodic with time T so that they maintain thescaling relationship. The proof for this is given in Appendix B. It can thebe shown that \u00b51\u00b52 = 1, which means that the general solution is given by(B.8):z(t) = \u00b5tT P1(t) + \u00b5\u2212 tT P2(t) (4.34)To demonstrate that this system has parametric resonance behavior, afew conditions must be met. In order for the action from which this wasderived to be accurate, the value of z(t) must remain small, else higher-order effects may come into play. This condition still allows one to studythe behavior when z is much less than the wavelength of the gravitons. Toextend this analysis, one can consider several modes:z\u00a8(t) = (A cos(\u03c91t) +B cos(\u03c92t))z(t) = f(t)z(t) (4.35)While more complicated than before, this is still periodic. Using theFloquet theory notation:x\u02d9(t) = P (t)x(t), P (t) =[0 1f(t) 0]x(t) =[zz\u02d9](4.36)The period T is given by f(t+T ) = f(t), which is true when \u03c91T = 2\u03c0n1and \u03c92T = 2\u03c0n2 where nj \u2208 Z. This can always be solved as long as \u03c9j \u2208 Rand \u03c9j \u0338= 0:2\u03c0n1\u03c91= 2\u03c0n2\u03c92\u2192 n1 = n2\u03c91\u03c92(4.37)The values of the frequencies must be such that there can always be twointegers n1 and n2 to satisfy this, and therefore some value of T . If a valueof T is chosen, a general function periodic in T can be written:f(t) =\u221e\u2211n=0An cos(2\u03c0ntT)(4.38)544.4. Instability in Geodesic ScatteringThe general solution can be found by using the boundary conditions andform of the B matrix as shown in (B.5) in a numerical simulation. For alonger period, the resulting eigenvalues of B would be much higher, meaningthe general solution will be proportional to some new variable \u03bb = T\u221a\u00b5 sothat the general solution will have a \u03bbt relationship. As T grows, so will\u00b5, and if it can be shown that \u03bb remains finite in the limit for a particularfunction, the exponential behavior would survive.As \u00b5 is generally found numerically, this approach is difficult to do in aprecise way, and thus it can only be conjectured that this may give a usefulsolution. One may argue that given that there is an exponential solution forany value of the frequency then, while a particular spectrum of frequencieswill affect the resulting rate of expansion, it does not eliminate the expo-nential dependence of the geodesic distance coming as a result of each modeindividually. To get a more analytical approach, stochastic gravity providesmore tools to make exact statements.4.4.1 Parametric Resonance from Stochastic GravityThere is a more direct way to find there is an accelerated expansion predictedby following the approach by Cho and Hu [70]. Starting from (4.24) andignoring the dissipation term, the form of the equation is:z\u00a8i(t) =2m\u03b4ik\u03bekl(t)zl(t) (4.39)This can be solved perturbatively by assuming some initial solution z0with z\u00a80 = 0:z\u00a8i1(t) =2m\u03b4ik\u03bekl(t)zl0(t) (4.40)z1(t) = z0(t) + \u03b4z1(t), \u03b4z1(t) =2m\u222b t0dt\u2032(t\u2212 t\u2032)\u03b4ik\u03bekl(t\u2032)zl0(t\u2032) (4.41)This is the point at which Cho and Hu stop. This is enough to find termsof the form\u221a\u27e8\u03b4z2\u27e9, and find the expected variation in the detector size.Continuing on from here, the next order can be found similarly:z2(t) = z1(t) + \u03b4z2(t), z\u00a8i2(t) =2m\u03b4ik\u03bekl(t)zl1(t) (4.42)2m\u03b4ik\u03bekl(t)zl0(t) + \u03b4\u00a8zi2(t) =2m\u03b4ik\u03bekl(t)zl0(t) +2m\u03b4ik\u03bekl(t)\u03b4zl1(t) (4.43)554.4. Instability in Geodesic Scattering\u03b4\u00a8zi2(t) =2m\u03b4ik\u03bekl(t)(2m\u222b t0dt\u2032(t\u2212 t\u2032)\u03b4lm\u03bemn(t\u2032)zn0 (t\u2032))=4m2\u03b4ik\u03b4lm\u222b t0dt\u2032(t\u2212 t\u2032)\u03bekl(t)\u03bemn(t\u2032)zn0 (t\u2032)(4.44)The stochastic tensors \u03be will fluctuate, and the fluctuations that occurare random, however their statistical properties are as usual described bythe noise kernel. Integrating over all possible configurations of the \u03be tensorsto get the expectation value of \u03b4\u00a8zi2(t):\u2329\u03b4\u00a8zi2(t)\u232a=\u222bD\u03be P [\u03be] ( 4m2\u03b4ik\u03b4lm\u222b t0dt\u2032(t\u2212 t\u2032)\u03bekl(t)\u03bemn(t\u2032)zn0 (t\u2032))=4m2\u03b4ik\u03b4lm\u222b t0dt\u2032(t\u2212 t\u2032)Nklmn(t, t\u2032)zn0 (t\u2032)(4.45)This is an average accelerated separation of the geodesics derived directlyfrom stochastic gravity. The noise kernel is semi-positive definite, and t \u2265 t\u2032,so the acceleration will be away from the origin - ie. an expansion. Notethat it depends on the noise kernel itself and is not based on anything suchas the regularization technique used. The approach used by Cho and Hudirectly leads to the prediction that geodesics should accelerate away fromeach other proportional to the initial separation and the noise kernel.The noise kernel as defined by Cho and Hu in this derivation is given by(eq. 28, 29 in [70]):Nklmn(t, t\u2032) =m2\u03ba216(2\u03c0)3d2dt2d2dt\u20322\u222bd3k d3k\u2032\u222bd3x d3x\u2032e\u2212ik\u20d7\u20d7\u02d9xe\u2212ik\u20d7\u2032\u20d7\u02d9x\u2032\u2211s\u03f5(s)kl (k\u20d7)\u03f5(s)nm(k\u20d7\u2032)G(1)(x, x\u2032)(4.46)G(1)(x, x\u2032) =\u2329{h(x), h(x\u2032)}\u232a (4.47)Substituting into (4.45) gives a cancellation of the mass and a scale of\u03ba2. This particular form is for the graviton noise. By including inducedfluctuations in the correlation function, one may also include the contribu-tion from fluctuating fields. The cancellation of the mass is as expected dueto the equivalence principle, while the remaining factor of \u03ba2 makes this asmall effect.564.5. Conclusion on Geodesic ScatteringStudying the higher order terms as I have done here was first done byBak et al. [73] maintaining the form of the noise kernel derived by Cho andHu. Their regularization technique involves a hard momentum cutoff, whichis justified by the finite size of the detector. In this approach, a negativeaverage geodesic separation was found, leading to a contraction of space overtime. This results in caustics where vacuum contracts into a single point ona timescale proportional to V230 , where V0 is the initial volume. This wouldsuggest that a large volume such as the universe should collapse very quickly- which is not what is observed.In principle (as explained in Chapter 3), the noise kernel should bepositive-semidefinite, however, the regularization technique used by Baket al. allows negative values. Determining whether this approach to cal-culating the noise kernel and subsequent regularization is correct may benecessary for understanding the effects on geodesics in stochastic gravity.The predictions of Bak et al. is similar to the Casimir effect [49] in whicha regularization technique to restrict the wavelengths between two platesleads to an inward force.If a regularization technique is instead used that maintains the positivesemi-definiteness of the noise kernel, an expansion should be expected in-stead. If one instead finds an outward expansion, the form of (4.39) suggestsa parametric oscillator leading to exponential expansion. Given the predic-tions of Bak et al., it may be more reasonable to accept the prediction thatthere is instead an expansion described by a strictly positive semi-definitenoise kernel. Calculations of this effect would necessitate regularization, andthus a technique that maintained the positive semi-definiteness of the noisekernel, while also respecting the physical limitations of the interferometersystem, may yield more sensible results.4.5 Conclusion on Geodesic ScatteringIn this chapter, I have demonstrated some approaches to understandingscattering due to fluctuations in stochastic gravity. I discussed some of thebackground on scattering such as multiple scattering and Rayleigh scatter-ing. It can be shown that the standard approaches to scattering in QFT donot allow for momentum to be transferred and allow elastic scattering. Someattempts at understanding the measurement of gravitons in interferometersmay have applications to this problem, particularly in the calculation ofgeodesic deviation due to fluctuations.There are some other approaches that could be considered. As the anal-574.5. Conclusion on Geodesic Scatteringogy to Rayleigh scattering suggests, an optical approach may be useful.In particular, one can consider a fluctuating dielectric corresponds with afluctuating index of refraction. This gives a relatively intuitive picture ofscattering as light will take the shortest time path, and fluctuations in thespeed of light in a medium will correspond to scattering in order to re-duce the time. An expanding and contracting spacetime will have regionsof faster and slower travel, and so similarly have scattering effects. Thismodel is complicated by time dependence, though this can be accounted forby simulations with a small time-step or by considering the problem in afour-dimensional Euclidian space such as through Wick rotation.While attempting to calculate this effect gives insight into the compat-ibility of QFT and GR, some of the results shown here can give an ideaof a very rough approach to calculating an approximation for the amountof scattering. The scattering in stochastic gravity is described by the noisekernel, which for fluctuations from a massless minimally-coupled field has aFourier transform (See 11.44 in [6]):N\u00b5\u03bd\u03b1\u03b2(p) =\u03b8(\u2212p2)2880\u03c0p4P\u00b5\u03bd\u03b1\u03b2 (4.48)Where P\u00b5\u03bd\u03b1\u03b2 is a projection operator. This shows that the scatteringwould have some relationship with the momentum, and therefore an inverserelationship with wavelength. This gives some motivation for expectinga similar behavior to Rayleigh scattering. If one makes the assumption -which is not well-founded in the single-scattering approach - that Rayleighscattering is an approximate description, it can be shown that it will stillbe negligible.The calculation of Rayleigh scattering assumes individual scatters of acertain volume, so an approximation may be done by assuming a discretespacetime where each point is a scattering sphere about the Planck scalein size, which is the length scale of the fluctuations, where each spacetimevolume along the path contributes to the scattering individually. A simplecalculation would be to assume a cross-section for each lattice point, thenassume extinction based on that cross-section - ie. there either is scatteringor there is not, leading to a drop in intensity.Taking the rough approximationA(\u03b8) = k2\u03b3V , where \u03b3 gives the strengthof the potential to take into account the strength of this gravitational inter-action, V = l3P for the size of the scatterers, and k2 \u223c \u03bb\u22122, one can calculatean approximate attenuation relationship with distance:584.5. Conclusion on Geodesic Scatteringd\u03c3d\u2126= |A(\u03b8)|2, \u03c3 \u223c \u03b32l6P\u03bb4(4.49)\u03a6 = \u03a60e\u2212n\u03c3z \u223c \u03a60e\u2212\u03b32 l3P\u03bb4z (4.50)Here, \u03a6 indicates the intensity, n is the scattering center density whichI have taken to be l\u22123P , and z is the distance. I will take z to be 100 billionlightyears so it is clear how this effect will appear over the scale of theentire universe. If one first looks at the range of wavelengths for light,such as around 1\u00b5m, one getsl3P\u03bb4z \u223c 10\u221254. This shows that the extremelysmall scales of the fluctuations give minuscule effects at relevant wavelengthsof light even without taking into account the factor of \u03b32 indicating theweakness of the gravitational field. Even waves on the Planck scale will bebarely affected even over such large distances.Assuming the Rayleigh scattering assumption is correct, even this veryrough approach shows negligible results. A multiple scattering approachassumes many small-angle scattering events over long distances to add upto noticeable angles, while extinction assumes that any scattering reducesthe intensity. It is, therefore, reasonable to assume that this approximationgives a result larger than if one instead calculated the multiple scattering.This demonstrates that stochastic gravity will have no measurable effect onthe scattering of waves, even on cosmological scales.While scattering suggested by Rayleigh scattering may not be an im-portant effect, effects of graviton noise on geodesics may be. Using theseparation of geodesics, one can infer the possibility for observable scatter-ing effects. Further study into calculating single-scattering and parametricresonance effects may be needed.59Chapter 5Pauli-Villars Regularization5.1 Issues with Breaking Lorentz InvarianceIn Chapter 3, I discussed the prediction of an apparent cosmological constantarising from a parametric resonance between quantum fields and the metricin stochastic gravity. The particular method used in that work was a high-energy cutoff, and it was shown how the predicted expansion rate relates tothe cutoff. It was suggested by Wang et al. [25] that it would be possibleto show the same thing in a Lorentz-invariant model. In particular, it wassuggested that Pauli-Villars could be used. Most generally, this argumentboiled down to demonstrating that:\u27e8\u03d5\u02d92\u27e9 \u223c \u039b4, \u27e8\u03d5\u00a82\u27e9 \u223c \u039b6 (5.1)It is apparent from dimensional analysis that this should be the case, andas a result, it may be reasonable to use Pauli-Villars instead of a high-energycutoff.Cree et al. [38] discussed further how the assumption that Lorentz in-variance is violated at high energy allows this solution for the cosmologicalconstant problem. Lorentz invariance mandates that \u27e8T\u00b5\u03bd\u27e9 \u223c g\u00b5\u03bd , and sothey argue violating Lorentz invariance removes the problem. This idea isstill complicated by the fact that the stress-energy tensor would then takethe form shown in (2.29).The cutoff is defined by taking some reference frame in which the mo-mentum in any direction is cut off at the same magnitude. This means thatthe result is inherently Lorentz non-invariant, given that it is defined for aparticular reference frame. The behavior of the cutoff can be understood byexamining it as the result of a discrete spacetime. As the spacetime expandswith a scale factor a, each discrete section is stretched to be a factor a larger,and thus the momentum is scaled with k \u2192 a\u22121k, meaning the cutoff scalesas a\u22121 as well. The stress-energy tensor will then scale with a\u22124. This isthe exact behavior expected for radiation.This interpretation complicates some of the interpretations of Wang et605.1. Issues with Breaking Lorentz Invarianceal. [25]. Cosmological models do not predict such a high radiation density.Furthermore, a vacuum energy density with the form of radiation wouldgrow very large at small scale factors. For the predictions of Wang et al. towork, this radiation must be explained away without explaining away thestochastic expansion.The important deviation from the standard approach to cosmology is todrop the assumptions of local homogeneity and isotropy. It is not clear if thesemiclassical result still holds if there are fluctuations, particularly if thesefluctuations are still on the order of the expectation value. An approach toresolving this was shown by Wang [39] by reformulating the cosmologicalconstant problem from a more general starting point. From this, the scalefactor can be extracted from its relationship to the metric determinant, andthe parametric resonance can be shown from there. As the variations existon such small scales, they can be shown to give the right behavior on largescales even for a large cosmological constant.This may mean that the assumptions about the vacuum in the semiclassi-cal analysis are incorrect. This would eliminate the effect of the expectationvalue, regardless of the regularization technique, at least in the way thesemiclassical approach suggests.There are still solutions to this problem without invoking any sort of re-formulation, though they come with some caveats. Imposing normal order-ing as suggested by Wald [14] could be one way to resolve this by separatelynormal ordering both stress-energy tensor operators:\u27e80| : T\u00b5\u03bd(x) : |0\u27e9 = 0, \u27e80| : T\u00b5\u03bd(x) :: T\u03b1\u03b2(y) : |0\u27e9 \u223c \u27e80| a2(a\u2020)2 |0\u27e9 (5.2)If one instead assumes a Lorentz-invariant regularization procedure, thesemiclassical result must be canceled in that approach as well. The normalordering approach may work here too in the exact same way, though thereis also the possibility of a cosmological constant counter-term defined by\u03b4\u03bbg\u00b5\u03bd = \u03ba \u27e8T\u00b5\u03bd\u27e9. This may have as much validity as counter-terms usedsuch as in dimensional regularization, given that the cosmological constantterm is an allowable term in the Einstein-Hilbert action. One would thenhave to impose some condition that the cosmological constant cancels theexpectation value of the stress-energy tensor of all fields.If one assumes the reformulation suggested by Wang [39] and use theEinstein-Langevin equation to expand out the stress-energy tensor, the quasiperi-odic function \u21262 takes a standard form of the Mathieu equation [25]:615.2. Introduction to Pauli-Villars\u21262 = (\u03ba \u27e8T00\u27e9+ \u03baa2( \u27e8T11\u27e9+ \u27e8T22\u27e9+ \u27e8T33\u27e9)+(\u03ba\u03be00+ \u03baa2(\u03be11+\u03be22+\u03be33)) (5.3)a\u00a8+\u21262a = a\u00a8+ \u03c920(1 + h cos \u03b3t)a = 0 (5.4)The introduction of a parametric resonance with zero expectation valuefor the stress-energy tensor requires a slightly different approach than theuse of the typical Mathieu equation. This is the case studied in section 4.4.Floquet theory still allows solutions in this form, and provided conditions aremet, Floquet theory still describes this system as unstable if the expectationvalue is zero. The form needed will depend on the assumptions made aboutthe formulation of the problem, though a parametric resonance is still asolution in either case.While the Lorentz-invariant cutoff may face some of the same issues asthe high-energy cutoff, it is still worth studying to determine if it can beused in a model that gives the parametric resonance behavior. In the rest ofthis chapter, I will demonstrate the formalism of Pauli-Villars regularizationin curved spacetime to examine if it could be used to impose a physicallymeaningful cutoff through Lorentz invariant regularization.5.2 Introduction to Pauli-VillarsPauli-Villars involves the subtraction of propagators to allow for a regular-ized one-loop integral. To find one-loop corrections to the propagator ofthis field, one can first see how the propagator will be divergent withoutregularization: \u222bddk(2\u03c0)dik2 +m2(5.5)Introducing one regulator, one can see that this will regularize this inte-gral:\u222bddk(2\u03c0)d(ik2 +m2\u2212 ik2 + \u039b2) =\u222bddk(2\u03c0)di(\u039b2 \u2212m2)(k2 +m2)(k2 + \u039b2)(5.6)This approach works based on this mathematical principle, however, ifone wants to demonstrate that some real physical process can have thisproperty then one must somehow define the action of the system so as toallow this form of the propagator. To demonstrate an example, one can lookat the case of a real scalar field \u03d5 with \u03d54 interactions in 2+1D.625.2. Introduction to Pauli-Villars5.2.1 Regularizing \u03d54 Theory in 2+1DFollowing the approach of \u2019t Hooft and Veltman [74] to define the lagrangianhere, I will introduce a second field \u03c8 so that the lagrangian is:L(0) = 12\u03d5(\u2212\u25a1\u2212m2)\u03d5\u2212 12\u03c8(\u2212\u25a1\u2212 \u039b2)\u03c8, L(I) = \u03bb4!(\u03d5+ i\u03c8)4 (5.7)To be consistent, I am following the principle here that every \u03c8 is mul-tiplied by a factor of i. This works because the real field still only has onedegree of freedom, and any observables will always involve an even numberof \u03c8 operators, so they will always remain real. The result is that the place-ment of the minus signs is such as to allow regularization. The propagatorswill still have the same sign on their own, but they will always in practice bemultiplied by \u22121 for the \u03c8 fields as a result. One can define the generatingfunctional including a source term:Z =\u222bD\u03d5D\u03c8 exp(iS(0) + iS(I) + i\u222bd4xJ(x)(\u03d5(x) + i\u03c8(x)))(5.8)The two-point correlation function is defined in the usual way by takingthe second derivative with respect to the source and sending it to zero:(1i2\u03b4\u03b4J(x)\u03b4\u03b4J(y)Z)\u2223\u2223\u2223\u2223J=0=\u222bD\u03d5D\u03c8eiS(0)+iS(I)(\u03d5(x) + i\u03c8(x))(\u03d5(y) + i\u03c8(y))(5.9)The free-field propagators can be labeled as D(x \u2212 y) for the positive-norm field and S(x\u2212 y) for the negative-norm field. These propagators aregiven by:D(x\u2212 y) =\u222bD\u03d5 eiS(0)\u03d5(x)\u03d5(y) =\u222bd3k(2\u03c0)3ieik(x\u2212y)k2 +m2(5.10)S(x\u2212 y) =\u222bD\u03c8 eiS(0)\u03c8(x)\u03c8(y) =\u222bd3k(2\u03c0)3ieik(x\u2212y)k2 + \u039b2(5.11)In the free theory, there is no interaction between the fields, so all corre-lation functions must either be reducible to these propagators using Wick\u2019stheorem or go to zero. To calculate a one-loop diagram, the first order in \u03bbis taken:635.2. Introduction to Pauli-Villars\u222bD\u03d5D\u03c8eiS(0)+iS(I)(\u03d5(x) + i\u03c8(x))(\u03d5(y) + i\u03c8(y))\u222bd3z(\u03d5(z) + i\u03c8(z))4=\u222bd3z [D(x\u2212 z)D(z \u2212 z)D(z \u2212 y)\u2212D(x\u2212 z)S(z \u2212 z)D(z \u2212 y)+ S(x\u2212 z)D(z \u2212 z)S(z \u2212 y)\u2212 S(x\u2212 z)S(z \u2212 z)S(z \u2212 y)]= \u2212i\u222bd3p d3k d3l(2\u03c0)9d3z eip(x\u2212z) eil(z\u2212y)[1(p2 +m2)(k2 +m2)(l2 +m2)\u2212 1(p2 +m2)(k2 + \u039b2)(l2 +m2)+1(p2 + \u039b2)(k2 +m2)(l2 + \u039b2)\u2212 1(p2 + \u039b2)(k2 + \u039b2)(l2 + \u039b2)]=\u222bd3p(2\u03c0)3eip(x\u2212y)(\u2212i(p2 +m2)2+\u2212i(p2 + \u039b2)2)[\u222bd3k(2\u03c0)3(1k2 +m2\u2212 1k2 + \u039b2)](5.12)Here I have used:\u222bd3l(2\u03c0)3d3z eip(x\u2212z) eil(z\u2212y) = \u03b43(p\u2212 l)eip(x\u2212y) (5.13)This result shows a convergent propagator part and an extra loop thatis independent of the incoming momentum. This loop is regularized by thePauli-Villars procedure in this case, and all regularization comes directlyfrom the form of the action. This demonstrates in principle that the actioncan be written in such a way as to allow regularization.The replacement of the propagator on its own appears like a mathemat-ical trick similar to the ones used in dimensional and zeta regularization.In the case of Pauli-Villars, there is, however, a very clear possibility fora physical interpretation in that the extra fields fit into the action muchlike a physical field would. Could the introduction of physical regulators bepossible, and what are the consequences, particularly in curved spacetime?645.3. Regularizing the Semiclassical Stress-Energy Tensor5.3 Regularizing the Semiclassical Stress-EnergyTensorFollowing the work of Anselmi [75], I will define Pauli-Villars regularizationusing a particular form of the DTNP Formalism [76]. In this formalism, theapproach to the Pauli-Villars regularization is chosen so as to respect certainsymmetries, in the original case BRST symmetry. This approach resemblesthe approach used in the previous section based on \u2019t Hooft and Veltman[74], though with a particular focus on coefficients. Here, going back to theminimally coupled real scalar field in curved spacetime but instead of onefield, I will introduce several massive real scalar fields in a 3+1D curvedspacetime with no self-interactions:L\u03d5 =N\u2211j=0(12cj\u221a\u2212gg\u00b5\u03bd\u2202\u00b5\u03d5j\u2202\u03bd\u03d5j \u2212 12cj\u221a\u2212gM2j \u03d52j ) (5.14)Here I have defined cj as a coefficient that multiplies the action of thescalar field \u03d5j with mass Mj . As a particular field is being regularized, Iwill define that field as the field \u03d50 with c0 = 1 and M0 = m. The standardPauli-Villars regularization is then just the case of N = 1 with c1 = \u22121 andM1 = \u039b.Here the fields are defined in a general way so that the fields can beinterpreted as in some sense physical, but defined so that mathematicalrelations between them can be defined in a systematic way. A source termis also needed: \u222bd4x\u221a\u2212g(x)J(x)(N\u2211j=0\u221acj\u03d5j(x)) (5.15)Fields are always multiplied by\u221acj so that the inclusion of these fieldsis done consistently, as with the placement of i in the previous example.This allows for the formal definition of propagators where required, thoughin calculating the stress-energy tensor there is no need for this term in thederivation, so it shall be omitted for simplicity. Following the standardprocedure for the semiclassical derivation of the stress-energy tensor usingthe effective potential:\u27e8T\u00b5\u03bd(x)\u27e9 = 2\u221a\u2212g(x) \u03b4\u0393\u03b4g\u00b5\u03bd(x) , ei\u0393 =\u222b N\u220fk=0D\u03d5k eiSk (5.16)655.3. Regularizing the Semiclassical Stress-Energy Tensor\u03b4\u0393\u03b4h\u00b5\u03bd(x)= \u2212ie\u2212i\u0393\u222b(N\u220fk=0D\u03d5k eiSk)N\u2211j=0(i\u03b4Sj\u03b4h\u00b5\u03bd(x))\u2248 e\u2212i\u0393N\u2211j=0\u222bD\u03d5j eiS(0)j\u03b4S(1)j\u03b4h\u00b5\u03bd(x)=N\u2211j=0\u27e8 \u03b4S(1)j\u03b4h\u00b5\u03bd(x)\u27e9(5.17)At the lowest order, the cross-terms between different fields can be ne-glected and so at the lowest order, there is a sum over the expectation valuesof the first-order terms of the action for each field in the free theory. Thefirst order term of the action for the field \u03d5j is given by:Sj = S(0)j + S(1)j +O(h2) =\u222bd4x (cj2\u03b7\u03b1\u03b2\u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u2212 cj2M2j \u03d5j(x)2)+\u222bd4x (cjF\u03b1\u03b2\u03c3\u03c1 \u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u2212 cj2F\u03c3\u03c1M2j \u03d5j(x)2)h\u03c3\u03c1(x) +O(h2)(5.18)N\u2211j=0\u27e8 \u03b4S(1)j\u03b4h\u00b5\u03bd(x)\u27e9 =N\u2211j=0(cjF\u03b1\u03b2\u00b5\u03bd \u27e8\u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u27e9 \u2212 cj2F\u00b5\u03bdM2j \u27e8\u03d5j(x)2\u27e9)(5.19)In the lowest order, the expectation values are taken in the free theory,meaning these expectation values come directly from the propagator:cj \u27e8\u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u27e9 =\u222bd4k(2\u03c0)4icjk\u03b1k\u03b2k2 +M2j(5.20)cjM2j \u27e8\u03d5j(x)2\u27e9 =\u222bd4k(2\u03c0)4icjM2jk2 +M2j(5.21)It is clear that both integrals are divergent, and the first integral is moreso. In this case, to be convergent there needs to be an overall power ofk8 in the denominator - meaning three regularization fields (N=3). Withthe single regularization field, a simple minus sign was enough to cancelany terms proportional to any powers of k in the numerator. In this case,however, a slightly more complex system of equations needs to be solved.Using some simplifications from symmetry:665.3. Regularizing the Semiclassical Stress-Energy TensorN\u2211j=0cj \u27e8\u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u27e9 = i\u03b7\u03b1\u03b24\u222bd4k(2\u03c0)4N\u2211j=0cjk2k2 +M2j(5.22)The principle is that one wants to eliminate the higher-order terms inthe numerator by setting the values of the coefficients such as to cancel theseterms. In (5.22), the sum over fractions should in principle still be divergent,as the integral over k has a power of 4 in the numerator - to be convergent, itmust be less than zero. Given N fields with N coefficients, the power in thenumerator can be decreased by a power of 2N by solving the correspondingN equations to eliminate the highest-order terms in the numerator. Each ofthe conditions can be written out in terms of an expansion:N\u2211j=0cjk2 +M2j=\u2211Nj=0A2Nk2N\u220fNj=0(k2 +M2j )=A0\u220fNj=0(k2 +M2j )(5.23)N\u2211j=0M2j cjk2 +M2j=\u2211Nj=0B2Nk2N\u220fNj=0(k2 +M2j )=B2k2 +B0\u220fNj=0(k2 +M2j )(5.24)I will now demonstrate the solution to this for the N = 3 case, with a fewsubstitutions. As stated, I will use c0 = 1, M0 = m, but I will also write theother terms as the largest mass at cutoff \u039b and two smaller or equal masses,given by M21 = \u03b1\u039b2, M22 = \u03b2\u039b2 and M23 = \u039b2, where 0 < \u03b1 < \u03b2 < 1. Inthis case, the conditions are given by the equations:A6 = 1 + c1 + c2 + c3 = 0 (5.25)A4 = m2(c1+c2+c3)+\u03b1\u039b2(1+c2+c3)+\u03b2\u039b2(1+c1+c3)+\u039b2(1+c1+c2) = 0(5.26)A2 = (\u03b1\u03b2 + \u03b1+ \u03b2)\u039b4 + c1(\u03b2m2\u039b2 +m2\u039b2 + \u03b2\u039b4)+ c2(\u03b1m2\u039b2 +m2\u039b2 + \u03b1\u039b4) + c3(\u03b1m2\u039b2 + \u03b2m2\u039b2 + \u03b1\u03b2\u039b4) = 0(5.27)A0 = \u03b1\u03b2\u039b6 + c1\u03b2m2\u039b4 + c2\u03b1m2\u039b4 + c3\u03b1\u03b2m2\u039b4 (5.28)As there are three coefficients and three conditions, then this system ofequations is solvable in principle. These conditions are significantly more675.3. Regularizing the Semiclassical Stress-Energy Tensorcomplicated due to the presence of a mass m. If it is assumed that \u039b >> mand that \u03b1 is large enough so thatM1 >> m, then the mass can be ignored asa small correction. Eliminating terms withm, simplifying using substitutionof other conditions, and factoring out \u039b from the conditions:A6 = 1 + c1 + c2 + c3 = 0, A4 = \u2212c1\u03b1\u2212 c2\u03b2 \u2212 c3 = 0 (5.29)A2 = (\u03b1\u03b2 + \u03b1+ \u03b2) + c1\u03b2 + c2\u03b1+ c3\u03b1\u03b2 = 0, A0 = \u03b1\u03b2\u039b6 (5.30)The first term is independent of the masses, and it is clear that in the caseof N = 1, there would be only one coefficient which must be \u22121, as is thecase in the example with one regularizing field. The remaining conditionsdemonstrate how the coefficients are locked in by the masses based on \u03b1 and\u03b2. Solving for the coefficients:c1 =\u2212\u03b2(\u03b2 \u2212 \u03b1)(1\u2212 \u03b1) , c2 =\u03b1(\u03b2 \u2212 \u03b1)(1\u2212 \u03b2) , c3 =\u2212\u03b1\u03b2(1\u2212 \u03b1)(1\u2212 \u03b2) (5.31)As long as \u03b1 < \u03b2 < 1, the signs of the coefficients can be determinedto be: c1 < 0, c2 > 0 and c3 < 0. This gives two negative fields and onepositive field, as opposed to the one negative field for N = 1. From (5.31), itis also clear that the coefficients will diverge as \u03b1\u2192 \u03b2 or \u03b2 \u2192 1. While thisappears like a condition on these values, this may be physically allowable inlimits that lead to convergent results, similar to the \u039b\u2192\u221e limit.These coefficients satisfy the conditions in (5.23), though (5.24) mustalso be satisfied. Writing out the values of Bj :B6 = m2 + c1\u03b1\u039b2 + c2\u03b2\u039b2 + c3\u039b2 = 0 (5.32)B4 = m2\u039b2(1 + \u03b1+ \u03b2) + c1\u03b1\u039b2(m2 + \u03b2\u039b2 + \u039b2)+ c2\u03b2\u039b2(m2 + \u03b1\u039b2 + \u039b2) + c3\u039b2(m2 + \u03b1\u039b2 + \u03b2\u039b2) = 0(5.33)B2 = (c1 + c2 + c3)\u03b1\u03b2\u039b6 + (1 + c2 + c3)\u03b2m2\u039b4+ (1 + c1 + c3)\u03b1m2\u039b4 + (1 + c1 + c2)\u03b1\u03b2m2\u039b4(5.34)B0 = (1 + c1 + c2 + c3)\u03b1\u03b2m2\u039b6 (5.35)685.3. Regularizing the Semiclassical Stress-Energy TensorIf the coefficients are the only variables fixed by the conditions, theconditions on B6 and B4 should be satisfied by the conditions on A6, A4and A2. Substituting in the solutions for the coefficients from the set ofconditions on the An factors, these conditions are automatically satisfied,even for m \u0338= 0. Substituting in the solutions for the coefficients, the onlynon-zero terms for m \u0338= 0 are:A0 = (\u03b1\u039b2 \u2212m2)(\u03b2\u039b2 \u2212m2)(\u039b2 \u2212m2), B2 = \u2212A0 (5.36)N\u2211j=0cj \u27e8\u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u27e9 = i\u03b7\u03b1\u03b24\u222bd4k(2\u03c0)4A0k2\u220fNj=0(k2 +M2j )(5.37)N\u2211j=0cjM2j \u27e8\u03d5j(x)2\u27e9 = i\u222bd4k(2\u03c0)4\u2212A0k2\u220fNj=0(k2 +M2j )(5.38)Substituting these expressions into the calculation gives massive simpli-fications:N\u2211j=0\u27e8 \u03b4S(1)j\u03b4h\u00b5\u03bd(x)\u27e9 =N\u2211j=0(cjF\u03b1\u03b2\u00b5\u03bd \u27e8\u2202\u03b1\u03d5j(x)\u2202\u03b2\u03d5j(x)\u27e9 \u2212 cj2F\u00b5\u03bdM2j \u27e8\u03d5j(x)2\u27e9)= (F\u03b1\u03b2\u00b5\u03bd i\u03b7\u03b1\u03b24+i2F\u00b5\u03bd)\u222bd4k(2\u03c0)4A0k2\u220fNj=0(k2 +M2j )=3iA08\u03b7\u00b5\u03bd\u222bd4k(2\u03c0)4k2\u220fNj=0(k2 +M2j )(5.39)The integral is convergent and completely factors out of the expression.Evaluating it requires the use of Feynman parameterization [77]:\u222bd4k(2\u03c0)4k2\u220fNj=0(k2 +M2j )=\u222b 10dx0...\u222b 10dxN\u222bd4k(2\u03c0)4N ! \u03b4(1\u2212\u2211Nj=0 xj)k2(\u2211Nj=0 xj(k2 \u2212M2j ))N+1(5.40)The momentum can be rearranged into a simplified form:695.3. Regularizing the Semiclassical Stress-Energy Tensor\u222bd4k(2\u03c0)4k2(k2 \u2212\u2206)N+1 =(\u22121)N2i(4\u03c0)2\u0393(N \u2212 2)\u0393(N + 1)\u22062\u2212N (5.41)This reiterates the requirement for the number of fields required for thisintegral as this is only convergent for N \u2212 2 > 0 due to the divergence ofthe gamma function, and so N = 3 is the minimum requirement. Finally,the form of this can be seen by substituting in to find \u27e8T\u00b5\u03bd\u27e9:\u27e8T\u00b5\u03bd\u27e9 =\u222b 10dx0...\u222b 10dxN2N ! \u03b4(1\u2212N\u2211j=0xj)3iA08\u03b7\u00b5\u03bd(\u22121)N2i(4\u03c0)2\u0393(N \u2212 2)\u0393(N + 1)\u22062\u2212N=38(4\u03c0)2\u03b7\u00b5\u03bd\u222b 10dx0...\u222b 10dxN\u03b4(1\u2212N\u2211j=0xj)(\u22121)N+1A0\u2206=3\u039b48(4\u03c0)2f(m,\u03b1, \u03b2)\u03b7\u00b5\u03bd(5.42)From this, an expression for the expectation value of the stress-energytensor is extracted. Clearly, it depends on the masses of all of the fields,with minimal restrictions on those masses. This expression will be verylarge for a large cutoff, just like the expectation value found through thehard cutoff method. This result gives an extra factor f(m,\u03b1, \u03b2) which maybe tuned so as to give a small or vanishing cosmological constant. While thepotential for a large cosmological constant is still present with this method,there is one convenience over the method used by Wang et al. [25]. In thehard cutoff case, the stress-energy tensor took the form as shown in equation(2.29), while in this case it is proportional to the metric.Examining this stress-energy tensor as a perfect fluid, given that it isproportional to the metric the equation of state is w = \u22121, correspondingto vacuum energy. By using Pauli-Villars, while the stress-energy tensor isstill divergent for \u039b\u2192\u221e, it now has a vacuum energy form, and the theoryis Lorentz-invariant.While this gives a solid foundation for this approach, there are sev-eral problems that I will study. First, I will look at interpreting the fieldsindividually, before continuing on to the second-order calculations of thestress-energy tensor required for study in stochastic gravity.705.4. Bare Ghost Fields in Curved Spacetime5.4 Bare Ghost Fields in Curved SpacetimeIn order to understand the issues with these negative-coefficient fields, I willlook at some of their properties when treated individually. While the regu-larization procedure keeps the fields together by how they are all present inthe source term, each field still has individual properties such as kinematicsregardless of the other fields. Ghost fields with negative coefficient have anegative energy and the potential for instability. Despite this, one can stilldefine the field so as to have the same form of the Feynman Propagator asa positive-norm field, depending on the method used to derive it. This isimportant for the regularization procedure I have used as the coefficient isassumed to multiply the propagator in order to allow the cancellation.5.4.1 Negative-norm in the Free-TheoryA field in the free-theory with coefficient c can be given by a lagrangian thatcan be solved for the equations of motion with the Euler-Lagrange equation:L = c2\u03b7\u00b5\u03bd\u2202\u00b5\u03d5\u2202\u03bd\u03d5\u2212 c2m2\u03d52 (5.43)\u2202\u00b5\u2202L\u2202(\u2202\u00b5\u03d5)\u2212 \u2202L\u2202\u03d5= c(\u03b7\u00b5\u03bd\u2202\u00b5\u2202\u03bd\u03d5\u2212m2\u03d5) = 0 (5.44)It is clear that in the free theory, the coefficient is irrelevant. This is tobe expected, as a field on its own will have the same kinematics regardlessof scaling by a coefficient. The coefficient will only become relevant wheninteracting with other fields. Despite this, one can still define the Hamil-tonian in such a way as to make the coefficient relevant. The relationshipwith the coefficient can be understood from the conjugate momentum:\u03c0 =\u2202L\u2202\u03d5\u02d9= \u2212c\u03d5\u02d9 (5.45)From the classical Hamiltonian field theory, one should expect that theconjugate momentum is proportional to c. Plugging this into the Hamilto-nian:H = \u2212c\u03d5\u02d92 \u2212 L = \u2212c\u03d5\u02d92 + c2\u03d5\u02d92 \u2212 c2\u03b4ij\u2202i\u03d5\u2202j\u03d5+c2m2\u03d52= c[\u221212\u03d5\u02d92 \u2212 12\u03b4ij\u2202i\u03d5\u2202j\u03d5+12m2\u03d52] = cH0(5.46)715.4. Bare Ghost Fields in Curved SpacetimeIt is clear that there is a scaling by a factor of c. The commutationrelations with the ladder operators are then also scaled by the coefficient:[H, a\u2020k] = c\u03c9ka\u2020k, [H, ak] = \u2212c\u03c9kak, \u03c9k =\u221a\u2223\u2223\u2223\u20d7k\u2223\u2223\u22232 +m2 (5.47)This shows that the ladder operators are affected by introducing thecoefficient. If the coefficient is negative, the creation operator will createmodes of momentum k\u20d7 but with a negative norm. Starting at the vacuumstate |0\u27e9, applying the creation operator decreases the energy, while theannihilation operator increases the energy. If the field acts on a positive-norm field, it will begin to lose energy as it feeds energy into the positive-norm field. As it is not bounded from below, and the positive-norm field isunbounded from above, the energy has the potential to feed between themdivergently. This seems unavoidable unless there is some sort of mechanismthat can bind the process so as to allow only finite energy transfer, such asan amplification effect or kinematic restrictions.5.4.2 Interaction with the Gravitational FieldThe bare ghost field needs some sort of interaction in order to feed its energy.Even if no other quantum fields are present, a gravitational coupling is allthat is needed to do this. I will use S-matrix formalism to calculate thelowest-order contribution from the two \u03d5 coupling with the transition fromthe vacuum to a state with outgoing graviton of momentum p\u20d7 and scalarswith momentum k\u20d7 and l\u20d7:\u27e8p; k, l| exp(i\u222bd4xLint)|0\u27e9 (5.48)This corresponds to a diagram like figure 5.1 a.To evaluate this in the lowest order, I will write the gravitational per-turbation in the transverse-traceless gauge. This can be written by writingit as a time-dependent sum over plane waves with momentum p\u20d7 with apolarization tensor and \u03f5(s)ij [46, 69, 70]:h\u03b1\u03b2(x) =\u222bd3p\u2032(2\u03c0)3\u2211sh(s)(p\u20d7\u2032, t)\u03f5(s)\u03b1\u03b2(p\u20d7\u2032)eip\u20d7\u2032\u00b7x\u20d7 (5.49)The gravitational perturbation using the approach I have used in thisthesis is classical, which this particular form allows. However, in the problem725.4. Bare Ghost Fields in Curved Spacetime(a) Interaction between a ghost field anda graviton(b) Pair creation effect between ghostfields and gravitonsFigure 5.1: Feynman diagrams of various ghost field interactions with gravi-tonsof particle creation I would like to study here, a quantum approach may beneeded in order to describe particle creation simply. There are several waysparticle creation can be calculated in semiclassical gravity. The idea is thata squeezed state is formed from a curved spacetime [13, 52]. In this case, Iwill treat the gravitons as quantum to demonstrate that in quantum gravity,this would be the way the runaway effect would work.In this form, the graviton is the sum of two scalar fields with different po-larizations [46]. This means that momentum can go into two different fields.The fields are treated separately in this interaction so that the contributionsto interactions with both fields are added, but there is no coupling involvingboth. This means that for the purposes here, one can focus on just onefield individually, though both fields will have energy put into them throughparticle creation. Focusing on just one polarization:\u27e8p|h\u03b1\u03b2(x) |0\u27e9 =\u222bd3p\u2032(2\u03c0)3\u27e8p|h(s)(p\u20d7\u2032, t) |0\u27e9 \u03f5(s)\u03b1\u03b2(p\u20d7\u2032)eip\u20d7\u2032\u00b7x\u20d7 (5.50)In this case, I will treat the metric perturbation as a field operator withladder operators b and b\u2020. Following through with this composition, thematrix element can be calculated in a straightforward way [69]:\u27e8p|h(s)(p\u20d7\u2032, t) |0\u27e9 = 1\u221a2m\u03c9p\u2032\u27e8p| b\u2020p\u20d7\u2032|0\u27e9 ei\u03c9p\u2032 t = (2\u03c0)3\u03b4(3)(p\u2212 p\u2032)\u221a2m\u03c9p\u2032ei\u03c9p\u2032 t (5.51)\u27e8p|h\u03b1\u03b2(x) |0\u27e9 = 1\u221a2m\u03c9pei\u03c9pt\u03f5(s)\u03b1\u03b2(p\u20d7)eip\u20d7\u00b7x\u20d7 (5.52)As used by Parikh et al. [69], the polarization tensor depends on thegauge, but can be chosen to be spatial and traceless - and must be perpen-735.4. Bare Ghost Fields in Curved Spacetimedicular to the momentum. These conditions follow directly from the gaugeconditions on h\u03b1\u03b2 which the polarization tensor must respect.\u03b4ij\u03f5ij(p\u20d7) = 0, pi\u03f5ij(p\u20d7) = 0 (5.53)Including this form as well as the fields in terms of the field operatorswill allow for an expansion in terms of ladder operators that will allow thecalculation of the matrix element. The field operators must also be expandedout into ladder operators in order to complete the matrix calculation. Somesimplifications are apparent, however:F\u03b1\u03b2h\u03b1\u03b2 = 12\u03b7\u03b1\u03b2h\u03b1\u03b2 \u223c \u03b4ij\u03f5ij = 0 (5.54)F\u00b5\u03bd\u03b1\u03b2h\u03b1\u03b2 = 14\u03b7\u03b1\u03b2\u03b7\u00b5\u03bdh\u03b1\u03b2 \u2212 12\u03b7\u03b1\u00b5\u03b7\u03b2\u03bdh\u03b1\u03b2 \u223c \u221212\u03f5\u00b5\u03bd (5.55)This simplifies the calculation significantly. Going back to the S-matrixelement and plugging this in:\u27e8p; k, l| exp(i\u222bd4xLint)|0\u27e9 \u223c \u27e8p; k, l| ic\u222bd4x \u2202\u00b5\u03d5(x)\u2202\u03bd\u03d5(x)h\u00b5\u03bd(x) |0\u27e9= ic\u222bdt\u222bd3x \u27e8k, l| \u2202i\u03d5(x)\u2202j\u03d5(x) |0\u27e9 \u03f5(s)ij(p\u20d7)\u221a2m\u03c9pei\u03c9pteip\u20d7\u00b7x\u20d7(5.56)The scalar field operator matrix elements can be calculated in a simi-lar way to the metric perturbation through direct substitution of the fieldoperator in terms of ladder operators:\u27e8k, l| \u2202i\u03d5(x)\u2202j\u03d5(x) |0\u27e9 =\u222bd3k\u2032d3l\u2032(2\u03c0)6\u27e8k, l| (ik\u2032iak\u2032e\u2212iEk\u2032 t+ik\u20d7\u2032\u00b7x\u20d7 \u2212 ik\u2032ia\u2020k\u2032eiEk\u2032 t\u2212ik\u20d7\u2032\u00b7x\u20d7)\u221a2Ek\u2032(il\u2032jal\u2032e\u2212iEl\u2032 t+il\u20d7\u2032\u00b7x\u20d7 \u2212 il\u2032ja\u2020l\u2032eiEl\u2032 t\u2212il\u20d7\u2032\u00b7x\u20d7)\u221a2El\u2032|0\u27e9=\u222bd3k\u2032d3l\u2032(2\u03c0)6\u2212k\u2032il\u2032jeiEk\u2032 t\u2212ik\u20d7\u2032\u00b7x\u20d7eiEl\u2032 t\u2212il\u20d7\u2032\u00b7x\u20d72\u221aEk\u2032El\u2032\u27e8k, l| a\u2020k\u2032a\u2020l\u2032 |0\u27e9=\u2212kiljeiEkt\u2212ik\u20d7\u00b7x\u20d7eiElt\u2212i\u20d7l\u00b7x\u20d72\u221aEkEl(5.57)745.4. Bare Ghost Fields in Curved SpacetimeSubstituting back in gives a relatively straightforward and familiar formof the S-matrix element to lowest order:\u27e8p; k, l| exp(i\u222bd4xLint)|0\u27e9\u223c ickilj\u222bdt\u222bd3xeiEkt\u2212ik\u20d7\u00b7x\u20d7eiElt\u2212i\u20d7l\u00b7x\u20d72\u221aEkEl\u03f5(s)ij(p\u20d7)\u221a2m\u03c9pei\u03c9pteip\u20d7\u00b7x\u20d7=\u2212ikilj(2\u03c0)62\u221a\u03c9k\u03c9l\u221a2m\u03c9p\u03f5(s)ij(k\u20d7 + l\u20d7) \u03b4(3)(p\u20d7\u2212 k\u20d7 \u2212 l\u20d7)\u03b4(Ek + El + \u03c9p)(5.58)This gives the scattering amplitude along with the delta functions formomentum and energy conservation which ensures that the total energyand momentum are conserved. Given that Ek = c\u03c9k, the only way for thisprocess to be physically allowable is if c < 0. Based on this result, for asingle negative-coefficient bare ghost field on its own, there will be divergentparticle creation as it is not bounded at high momentum.It is clear that in order for the negative-norm fields to be allowable, theremust be some sort of regularization of this process on top of the propagatorregularization. This expression can be used to demonstrate the scatteringamplitude for a negative-energy solution, however, a slight difference comeswith a positive-energy solution, as the gravitons will be consumed by particlecreation instead of created with the particles:\u27e80|h\u03b1\u03b2(x) |p\u27e9 = 1\u221a2m\u03c9pe\u2212i\u03c9pt\u03f5(s)\u03b1\u03b2(p\u20d7)eip\u20d7\u00b7x\u20d7 (5.59)\u27e80; k, l| exp(i\u222bd4xLint)|p; 0\u27e9\u223c \u2212ikilj(2\u03c0)62\u221a\u03c9k\u03c9l\u221a2m\u03c9p\u03f5(s)ij(p\u20d7) \u03b4(3)(p\u20d7\u2212 k\u20d7 \u2212 l\u20d7)\u03b4(Ek + El \u2212 \u03c9p)(5.60)Given that there are gravitons being created by the negative-norm fieldsand annihilated by the positive-norm fields, there must either be some bal-ance, or there must be kinematic restrictions. The amplitude will be inde-pendent of the coefficient because it cancels due to Ek = c\u03c9k in the denom-inator. Only the energy and momentum conservation will take into accountthe coefficients. The rate will then depend in part on the momentum asthe amplitude grows roughly with\u221ak, and the size of the momentum phasespace for each field.755.4. Bare Ghost Fields in Curved SpacetimeThe momentum conservation sets p\u20d7 = k\u20d7 + l\u20d7, leaving the momentumcondition c\u03c9k + c\u03c9l + \u03c9p for negative-norm fields and c\u03c9k + c\u03c9l \u2212 \u03c9p forpositive-norm fields. The situations are mirror copies for equal magnitudesof c with opposite signs as the sign on \u03c9p cancels. The relevant propertywill therefore be only the relative magnitudes of the coefficient between thefields and their masses. There is a massive simplification to be made ifthe polarization tensor is taken into account. If a particular form of thepolarization tensor is assumed, and use p\u20d7 and k\u20d7 as the free vectors withl\u20d7 = p\u20d7\u2212 k\u20d7, the form of expressions is:\u2212ikilj(2\u03c0)62\u221a\u03c9k\u03c9l\u221a2m\u03c9p\u03f5(+)ij(p\u20d7) \u03b4(c\u03c9k+c\u03c9l\u2212\u03c9p) =i(k2x \u2212 k2y)(2\u03c0)62\u221a\u03c9k\u03c9l\u221a2m\u03c9p\u03b4(c\u03c9k+c\u03c9l\u2212\u03c9p)(5.61)Here I have substituted for l\u20d7, used the polarization tensor conditions, andchosen to use a positive polarization as an example. Now one can considerfor a given value of p\u20d7 the space of allowable momentum k\u20d7 and the relativecontribution to the scattering amplitude. This can be written in a standardscattering amplitude form:P (c, p\u20d7) \u223c\u222bd3k(2\u03c0)3(k2x \u2212 k2y)2m\u03c9k\u03c9l\u03c9p\u03b4(c\u03c9k + c\u03c9l \u2212 \u03c9p) (5.62)I have removed the constants to keep the relevant quantities for com-paring scattering amplitudes. The next step is to carry out the momentumintegral over k\u20d7 for a given p\u20d7 that obeys the kinematics for a given c.5.4.3 Scattering Amplitudes of Different CoefficientsStarting from p\u20d7 = k\u20d7+ l\u20d7 and c\u03c9k+c\u03c9l = \u03c9p, some different restrictions can beplaced on the resulting allowable momenta. The case of c = 1 is the standardcase of scattering that may come from decay from one particle to two. For amassless decaying particle, this is not physically allowable without some sortof fourth particle, such as nuclear recoil, or, as I will discuss, a higher-orderFeynman diagram in the case of pair creation in QED.In the case of this momentum integral, it will be simplest to use carte-sian components. I will consider p\u20d7 = (0, 0, p), k\u20d7 = (kx, ky, kz) and l\u20d7 =(\u2212kx,\u2212ky, p\u2212 kz). The effect of the x and y components is to give a largereffective mass in the energy so that the constraint takes the form:c\u221ak2z + (k2x + k2y +m2) + c\u221a(p\u2212 kz)2 + (k2x + k2y +m2) = p (5.63)765.4. Bare Ghost Fields in Curved SpacetimeThis can be further simplified by making the kx and ky polar coordinatesso that kx = \u03be cos(\u03d5) and ky = \u03be sin(\u03d5), leaving an effective mass m2eff =\u03be2+m2. This allows the angle \u03d5 to be integrated out as the only dependencewill be in the numerator, and it will have no restrictions and will end up asa multiplicative constant of \u03c0:\u222b 2\u03c00(cos2(\u03d5)\u2212 sin2(\u03d5))2d\u03d5 = \u03c0 (5.64)There are now two degrees of freedom left; kz and the effective mass \u03be.Solving for p given an effective mass illuminates the space of solutions:c\u221ak2z +m2eff + c\u221a(p\u2212 kz)2 +m2eff = p, p =2c (\u221ak2z +m2eff \u2212 kz)1\u2212 c2(5.65)This will only have positive-definite solutions for |c| < 1, meaning thisprocess is not allowable for fields with coefficients with |c| \u2265 1. The coeffi-cient requirements were that 1+c1+c2+c3 = 0 with two negative coefficientsand one positive coefficient, based on the order of the masses. The field withc = 1 will not contribute to this process, meaning the other three fields maycontribute. As long as certain values are picked for the parameters \u03b1 and \u03b2,it is possible to have no fields with coefficients with a magnitude less than1. This introduces a slight extra condition in that for \u03b1 < 12 , one must alsoobey 1 \u2212 \u03b1 < \u03b2, but with this condition, one will never have a coefficientwith a magnitude less than one. As long as it is not physically allowableto have single-graviton interactions, there is no issue with runaway particlecreation from this process.This calculation was done at lowest order, however, bringing it up tohigher order removes this restriction. For example, a diagram for pair cre-ation (similar to electron-positron pair creation in QED) with two gravitonslifts this restriction. In this case, an example of a diagram and correspondingS-matrix element can be given by:\u27e8p, q; k, l| exp(i\u222bd4xLint)|0\u27e9 (5.66)This corresponds to a diagram like figure 5.1 b.Ignoring the various other terms in this element, including those withsecond-order contributions in graviton self-interaction and second-order scalar-graviton vertices, this expression demonstrates that kinematic restrictionscan be lifted, leaving a new set of conditions:775.4. Bare Ghost Fields in Curved Spacetimek\u20d7 + l\u20d7 + p\u20d7+ q\u20d7 = 0, c\u03c9k + c\u03c9l + \u03c9p + \u03c9q = 0 (5.67)This, along with the higher-order diagrams, introduces even more kine-matic restrictions on the ghost fields that aren\u2019t as easily explained away. Acareful balance must be introduced between the positive and negative coeffi-cient fields to cancel these diagrams. A similar result appears when lookingdirectly at scalar-scalar scattering through graviton exchange.5.4.4 Scalar Field ScatteringStudying the interaction between the gravitational field and ghost fieldsshowed that there are kinematic restrictions on scalar-graviton interactionsdue to the conservation of energy and momentum. In that case, the gravitonis not strictly considered quantum and is \u2019on-shell\u2019; it is a mode generatedin a classical field. To consider interactions between scalar fields that aremediated by the gravitational field, the kinematic restrictions can be relaxedif the gravitons are taken as \u2019off-shell\u2019 - implying some sort of quantum grav-itational interaction. In this case, the kinematics of such an interaction is ofinterest, even if the validity of such a physical process may be questionable.For a field with coefficients c1 and momenta k\u20d7 and l\u20d7 interacting with afield with coefficient c2 and momenta p\u20d7 and q\u20d7, the conservation of momentumand energy are given by:k\u20d7 + l\u20d7 = p\u20d7+ q\u20d7, c1(\u03c9k + \u03c9l) + c2(\u03c9p + \u03c9q) = 0 (5.68)When the signs of the coefficients are opposite, energy can be fed fromthe negative-norm field to the positive-norm field as the former acts like areservoir with infinite energy.This has been pointed out as a clear issue with the ghost fields [78],however, this is usually only discussed in the context of fields with equaland opposite coefficients. As seen when dealing with the graviton case,the relative magnitude of the coefficients matters in the kinematics. Thereare systems with negative norm but different coefficients that can displayamplification behavior [79] instead of completely divergent behavior. In thecase with four fields, there will be two positive and two negative coefficientfields, meaning four interactions like this.I have outlined a rough approach to the problem, though the specifics ofthe constraints must be worked out. It may be that introducing more fieldsmay help at higher orders so that there are extra degrees of freedom to beconstrained. At arbitrary order one may have to introduce arbitrarily many785.5. The Stress-Energy Tensor in Stochastic Gravityfields, running into the usual problem with non-renormalizability in studyinggravity. The possibility of amplification may be a worthwhile investigationin this case, given that there may be a more complicated interplay betweenthe fields than just divergent particle creation.5.5 The Stress-Energy Tensor in StochasticGravityLooking at the kinematics of the ghost fields that appear in Pauli-Villarsregularization when gravity is involved demonstrates some issues with tak-ing the technique to describe a physical system beyond the regularizationprocedure itself. On top of this, one can show that there are issues withPauli-Villars when trying to regularize processes in stochastic gravity.In stochastic gravity, the stress-energy tensor has a stochastic part t\u00b5\u03bdwhich has fluctuations dictated by the noise kernel. The noise kernel can becalculated by considering the expectation value of the stress-energy tensorand its two-point correlation function. The calculation can be done in thein-out formalism. In order to use Pauli-Villars, I will first reintroduce theLagrangian in (5.14) and expand it to first order in the metric perturbation:L\u03d5 =N\u2211j=0(12cj\u221a\u2212gg\u00b5\u03bd\u2202\u00b5\u03d5j\u2202\u03bd\u03d5j \u2212 12cj\u221a\u2212gM2j \u03d52j )=N\u2211j=0(12cj\u03b7\u00b5\u03bd\u2202\u00b5\u03d5j\u2202\u03bd\u03d5j \u2212 12cjM2j \u03d52j )+N\u2211j=0(cjF\u00b5\u03bd\u03b1\u03b2\u2202\u00b5\u03d5j\u2202\u03bd\u03d5j \u2212 12cjF\u03b1\u03b2M2j \u03d52j )h\u03b1\u03b2(x) +O(h2)(5.69)This expression separates into the Minkowski free theory for each field,plus separate interaction terms with the metric perturbations. As I willshow, this particular form runs into issues fairly quickly in this formulationdue to the lack of interaction between the different fields. The noise kernelcan be found expanded around Minkowski spacetime:T\u03b1\u03b2(x) = 2\u03b4S\u03d5\u03b4h\u03b1\u03b2(x)(5.70)795.5. The Stress-Energy Tensor in Stochastic Gravity12\u27e8{T\u03b1\u03b2(x), T \u03c3\u03c1(y)}\u27e9 = 2e\u2212i\u0393\u222b \u220fjD\u03d5jeiSj{(\u2211n\u03b4Sn\u03b4h\u03b1\u03b2(x)), (\u2211m\u03b4Sm\u03b4h\u03c3\u03c1(y))}= 2F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1 \u27e8{(\u2211ncn\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5n(x)), (\u2211m\u2202\u03b3\u03d5m(y)\u2202\u03f5\u03d5m(y))}\u27e9= 4F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1\u2211n(\u2211m\u0338=ncncm \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5n(x)\u27e9 \u27e8\u2202\u03b3\u03d5m(y)\u2202\u03f5\u03d5m(y)\u27e9+ cncm \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5n(x)\u2202\u03b3\u03d5m(y)\u2202\u03f5\u03d5m(y)\u27e9)(5.71)To get to the noise kernel from this, the product of the expectationvalues should be subtracted. These were calculated before and notably, thecontribution from each field was added together:\u27e8T\u03b1\u03b2(x)\u27e9 \u27e8T \u03c3\u03c1(y)\u27e9= 4F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1 \u27e8\u2211ncn\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5n(x)\u27e9 \u27e8\u2211m\u2202\u03b3\u03d5m(y)\u2202\u03f5\u03d5m(y)\u27e9 (5.72)It is clear immediately that the terms with different n and m will cancel.Furthermore, when n and m are equal one will also lose the terms wherethere are two separate loops at x and y instead of two \u03d5 loop terms eachmade of two propagators cycling between x and y. After subtracting this,all that remains are the terms:12\u27e8{t\u03b1\u03b2(x), t\u03c3\u03c1(y)}\u27e9= 4F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1\u2211nc2n( \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03b3\u03d5n(y)\u27e9 \u27e8\u2202\u03bd\u03d5n(x)\u2202\u03f5\u03d5n(y)\u27e9+ \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03f5\u03d5n(y)\u27e9 \u27e8\u2202\u03bd\u03d5n(x)\u2202\u03b3\u03d5n(y)\u27e9)(5.73)This clearly has issues with regularization. As the cn are real, theirsquare will be positive-definite, so no cancellation can happen. The diagramsthis corresponds to are the sum over loops with different fields, but thismeans that the propagators themselves are not summed over as in figure 5.3.Instead, two propagators for each field are multiplied, which would requirecross-terms to recover when substituting in sums over the propagators.This is the reason for the success of Pauli-Villars in regularizing theexpectation value; it is an accident of the fact that it is proportional to thesum over propagators in the first order, as shown in figure 5.2.805.5. The Stress-Energy Tensor in Stochastic Gravity(a) Positive-norm field contribution (b) Negative-norm field contributionFigure 5.2: Feynman diagrams contributing to the stress-energy tensor ex-pectation value for positive and negative fields.(a) Positive-norm field contribution (b) Negative-norm field contributionFigure 5.3: Feynman diagrams contributing to the noise kernel for positiveand negative fields.5.5.1 Modifying the LagrangianTo fix this in the perturbation theory, interaction terms between differentfields in the first-order interaction term would need to be introduced:L(1) =N\u2211i=0N\u2211j=0\u221acicj(F\u00b5\u03bd\u03b1\u03b2\u2202\u00b5\u03d5i\u2202\u03bd\u03d5j \u2212 12F\u03b1\u03b2MiMj\u03d5i\u03d5j)h\u03b1\u03b2(x) +O(h2)(5.74)However, this does not make much physical sense. This expression comesfrom expanding the action in terms of\u221a\u2212g and g\u00b5\u03bd , which means it can\u2019t beso easily split up. General relativity does not assume there is a backgroundto be respected while separating out these background-dependent terms doesnot respect this. If a similar reformulation is done to the free-theory term,self-interaction terms at zeroth order in the perturbations would be includedwhich makes the theory in Minkowski spacetime different from the usualnon-interacting free theory. This is a clear violation of the principles of GR,however, I will demonstrate that even this does not go far enough to solvethe problem.815.5. The Stress-Energy Tensor in Stochastic GravityThe reason for this modification is to introduce a useful relationship. Asthe free-theory Lagrangian is used for calculating the expectation values,all interaction between fields is eliminated at this level. This allows thetwo-point correlation functions to have an implicit delta function betweenfields.As a demonstration, ignoring the mass term, the noise kernel is given bycalculating:12\u27e8{T\u03b1\u03b2(x), T \u03c3\u03c1(y)}\u27e9 = 2e\u2212i\u0393\u222b \u220fjD\u03d5jeiS\u03d5{ \u03b4S\u03d5\u03b4h\u03b1\u03b2(x),\u03b4S\u03d5\u03b4h\u03c3\u03c1(y)}= 4F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1\u2211n,m,k,l\u221acncmckcl \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5m(x)\u2202\u03b3\u03d5k(y)\u2202\u03f5\u03d5l(y)\u27e9(5.75)As with the expectation value, the free-theory propagators force fieldsto be paired up. In this case, that allows two propagators of different fieldsto be paired up, which is a weaker constraint than in the previous section:4F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1\u2211n,m,k,l\u221acncmckcl \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5m(x)\u2202\u03b3\u03d5k(y)\u2202\u03f5\u03d5l(y)\u27e9= 4F\u00b5\u03bd\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1\u2211n,mcncm( \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5n(x)\u27e9 \u27e8\u2202\u03b3\u03d5m(y)\u2202\u03f5\u03d5m(y)\u27e9+ \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03b3\u03d5n(y)\u27e9 \u27e8\u2202\u03bd\u03d5m(x)\u2202\u03f5\u03d5m(y)\u27e9+ \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03f5\u03d5n(y)\u27e9 \u27e8\u2202\u03bd\u03d5m(x)\u2202\u03b3\u03d5m(y)\u27e9)(5.76)As before, the first term is canceled exactly by subtracting the squaredexpectation value. The remaining expression is now regularizable by tuningthe coefficients:\u2211ncn \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03b3\u03d5n(y)\u27e9\u2211mcm \u27e8\u2202\u03bd\u03d5m(x)\u2202\u03f5\u03d5m(y)\u27e9+ (\u03b3 \u2190\u2192 \u03f5) (5.77)This demonstrates that this approach could work for the terms not de-pendent on mass. However, if one reintroduces masses, there will be newcross-terms:825.6. Conclusion on Pauli-Villars\u2212 2(F\u00b5\u03bd\u03b1\u03b2F\u03c3\u03c1\u2211n,m,k,l\u221acncmckclMkMl \u27e8\u2202\u00b5\u03d5n(x)\u2202\u03bd\u03d5m(x)\u03d5k(y)\u03d5l(y)\u27e9+ F\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1\u2211n,m,k,l\u221acncmckclMnMm \u27e8\u03d5n(x)\u03d5m(x)\u2202\u03b3\u03d5k(y)\u2202\u03f5\u03d5l(y)\u27e9)(5.78)When applying Wick\u2019s theorem to bring out the two-point correlationfunctions, there will in general be terms proportional to masses of two dif-ferent fields. For example, one of those terms will be:\u22122\u2211n,mcncmMnMm(F\u00b5\u03bd\u03b1\u03b2F\u03c3\u03c1 \u27e8\u2202\u00b5\u03d5n(x)\u03d5n(y)\u27e9 \u27e8\u2202\u03bd\u03d5m(x)\u03d5m(y)\u27e9+ F\u03b1\u03b2F\u03b3\u03f5\u03c3\u03c1 \u27e8\u03d5n(x)\u2202\u03b3\u03d5n(y)\u27e9 \u27e8\u03d5m(x)\u2202\u03f5\u03d5m(y)\u27e9)(5.79)This expression can not be factored into the free theory propagatorslike the kinetic terms could. The inability to factor the propagator termsproportional to masses means it will not work to factor the Pauli-Villarsfields.This serves as a demonstration of the issue of defining an action thatallows for the summing over propagators as required in Pauli-Villars. In thiscase, the necessary inclusion of mass terms causes issues. An approach thatrespects Lorentz-invariance, background independence, and real observableswould appear to require even more convoluted conditions.5.6 Conclusion on Pauli-VillarsIn order to use a Lorentz-invariant model of vacuum fluctuations, a regular-ization technique that respects such a symmetry must be used. Pauli-Villarssatisfies this condition and introduces a cutoff, though otherwise comes withmany issues. If the expectation value of the stress-energy tensor is ignoredor the cosmological constant problem is reformulated so as to neglect it, theresult of a parametric resonance predicted by Wang et al. [25] should still bevalid, even if the result is slightly modified. As Lorentz-invariance appearsto be a property worth maintaining, this model is worth investigating.Pauli-Villars can be introduced in a formal way using real coefficients forfields of various masses. In this model, it is assumed that there is a massless(or low mass) field being regularized by high mass Pauli-Villars fields. This835.6. Conclusion on Pauli-Villarsmethod can be used for lowest-order calculations of the expectation value,however, higher-order contributions including those used to calculate thenoise kernel run into issues. Trying to redefine the formalism forms newproblems under the assumptions that the coefficients are constrained by thefacts that they must give real values for observables and obey backgroundindependence.Individually, the fields must interact with the gravitational field. Allow-ing interactions with gravitons, one can show extra kinematic conditions onthe fields that further restrict the coefficients. The system of Pauli-Villarsfields interacting through virtual graviton exchange may be worth some in-vestigation, particularly in the context of amplification. Overall, given thefactors discussed in this chapter, Pauli-Villars regularization does not appearto be a useful approach in stochastic gravity.84Chapter 6Conclusion and Future WorkIn the last chapter, I discussed the motivation for a Lorentz-invariant modelof vacuum fluctuations to describe the accelerated expansion of the universeas described by Wang et al. [25]. Due to various factors, a model basedon Pauli-Villars regularization faces some issues in describing a physicalsystem. In this chapter, I will re-examine the approach of Wang et al. whileassuming Lorentz-invariance.6.1 The Lorentz-Invariant ApproachStarting with the same metric as Wang et al. [25], given in (3.40), andfollowing through the analysis to (3.41), everything is the same. In this case,using a massless scalar field, one can write out the stress-energy tensor:T\u00b5\u03bd = \u2202\u00b5\u03d5\u2202\u03bd\u03d5\u2212 12g\u00b5\u03bd\u2202\u03bb\u03d5\u2202\u03bb\u03d5 (6.1)Up to this point, all of the analysis to get this result is consistent withLorentz invariance. The difference comes in the calculation of \u21262(x\u20d7, t). Tofind \u21262 one can simply substitute the stress-energy tensor using the givenmetric:T00 = \u03d5\u02d92 +12(\u2212\u03d5\u02d92 + a\u22122\u22072\u03d5) (6.2)Tii = (\u2202i\u03d5)2 \u2212 a22(\u2212\u03d5\u02d92 + a\u22122\u22072\u03d5) (6.3)T00+1a2(T11+T22+T33) = \u03d5\u02d92\u221212\u03d5\u02d92+12a2\u22072\u03d5+ 1a2\u22072\u03d5+32\u03d5\u02d92\u2212 32a2\u22072\u03d5 = 2\u03d5\u02d92(6.4)With the hard cutoff method, one can evaluate \u27e8\u21262\u27e9 and \u27e8(d\u2126dt )2\u27e9 directlyto find the form of the parametric oscillator, as is done by Wang et al. Thereis nothing inherently Lorentz-noninvariant up to this point, however, thecalculation of \u2126 requires the selection of a regularization technique.856.2. Approaches Without a Cutoff6.1.1 Using the Einstein-Langevin EquationIn the calculation methods used in this thesis, the starting point of \u03d5\u02d92 wouldnot be seen in a Lorentz-invariant approach. It requires taking the expec-tation value of this result, which is not straightforward in Lorentz-invarianttheory based in functional formalism. The perturbation theory assumesa small perturbation, but the parametric resonance does not assume this.Perturbation theory would be valid in the range where a2 << 1, which it isnot assumed to be in this case, particularly given that an expansion aroundMinkowski spacetime is taken at a2 = 1.Therefore, I would suggest taking a different approach to this calculationby making a few assumptions discussed previously. First, I would suggestnormal ordering of the stress-energy tensor to remove all contributions fromthe expectation value. From this point, using the Einstein-Langevin equa-tion directly relates the stochastic component to \u21262, such as in (5.3). If onetakes \u27e8\u03be\u00b5\u03bd\u27e9 = 0, this takes the form:\u21262 = \u03ba\u03be00 +\u03baa2(\u03be11 + \u03be22 + \u03be33) (6.5)As \u27e8\u03be\u00b5\u03bd\u27e9 = 0, this will exhibit oscillatory behavior based on the Fouriertransform of the noise kernel. Given zero expectation value, this system hasthe same sort of behavior as the form of the Mathieu equation discussed inChapter 4, such as (4.31). This oscillator is different from the one studiedby Wang et al., however, it has some simplifications in that the vacuumexpectation value is zero in Minkowski spacetime as prescribed by Wald\u2019saxioms [14]. The comparison of this approach with the one used in Chapter4 and the one used by Wang et al. may offer an alternative that allows aLorentz-invariant approach.6.2 Approaches Without a CutoffRegardless of taking either the approach of Wang et al. [25] or a Lorentz-invariant approach as I have described, a regularization technique must beused. The cutoff appears to remain important in the stochastic gravityapproach to the cosmological constant problem regardless of the presenceof the expectation value \u27e8T\u00b5\u03bd\u27e9. There may, however, be an alternativeapproach to gain a parameter that describes the vacuum fluctuations.The last approach I will suggest is to not use a cutoff at all. In thePauli-Villars approach, extra massive fields were included with negative co-efficients. In reality, it\u2019s likely that there are massive fields that modern866.2. Approaches Without a Cutoffexperiments are not able to measure. If one takes a massive field and usesa different regularization procedure without a cutoff - particularly dimen-sional regularization - one gets a similar result. Starting from a first-ordercontribution to the stress-energy tensor for a massive scalar field with mass\u039b in d = 4\u2212 \u03f5 dimensions [77]:\u27e8T\u00b5\u03bd(x)\u27e9 = F\u00b5\u03bd\u03b1\u03b2 \u27e8\u2202\u03b1\u03d5(x)\u2202\u03b2\u03d5(x)\u27e9 \u2212 12F\u00b5\u03bd\u039b2 \u27e8\u03d5(x)2\u27e9 (6.6)\u27e8\u03d5(x)2\u27e9 = \u00b5\u03f5\u222bddk(2\u03c0)dik2 \u2212 \u039b2 + i\u03f5 =1(4\u03c0)d2\u0393(1\u2212 d2)(\u039b2)d2\u22121=\u039b2(4\u03c0)2\u0393(\u22121 + \u03f52)(\u221a4\u03c0\u00b5\u039b)\u03f5=\u039b2(4\u03c0)2(\u22122\u03f5\u2212 2 log(\u00b5\u039b)\u2212 log(4\u03c0) + \u03b3 \u2212 1 +O(\u03f5))(6.7)\u27e8\u2202\u03b1\u03d5(x)\u2202\u03b2\u03d5(x)\u27e9 = \u00b5\u03f5\u222bddk(2\u03c0)dik\u03b1k\u03b2k2 \u2212 \u039b2 + i\u03f5 =\u22121(4\u03c0)d2\u03b7\u03b1\u03b22\u0393(\u2212d2)(\u039b2)d2=\u2212\u039b4\u03b7\u03b1\u03b22(4\u03c0)2\u0393(\u22122 + \u03f52)(\u221a4\u03c0\u00b5\u039b)\u03f5=\u2212\u039b4\u03b7\u03b1\u03b22(4\u03c0)2(1\u03f5+ log(\u00b5\u039b)+12log(4\u03c0)\u2212 \u03b32+34+O(\u03f5))(6.8)Taking the limit as \u03f5 \u2192 0, canceling the divergent expressions usingcounterterms and using the MS subtraction scheme gives convergent ex-pressions with a logarithmic part. The stress-energy tensor, therefore, endsup with a familiar form, except with a logarithmic dependence on the massand scale:\u27e8T\u00b5\u03bd(x)\u27e9 = \u2212F\u00b5\u03bd\u03b1\u03b2 \u039b4\u03b7\u03b1\u03b22(4\u03c0)2log(\u00b5\u039b)+ F\u00b5\u03bd \u039b4(4\u03c0)2log(\u00b5\u039b)=\u039b4\u03b7\u00b5\u03bd4(4\u03c0)2log(\u00b5\u039b)(6.9)This expression is very similar to the result from Pauli-Villars regulariza-tion, except with a modification based on some scale \u00b5. This is one aspectthat is not present in Pauli-Villars regularization and is usually attributedto an experimental scale. This is problematic when it comes to interpreting876.3. Future workthe result in a cosmological context which should behave on a universal scaleindependent of any experiment.The interpretation of this is somewhat different than the massless scalarfield included in the work by Wang et al. [25]. In that case, the field wasinterpreted as massless and a cutoff is included in regularization. In boththe Pauli-Villars case and in the case where a massive field is studied usingdimensional regularization, a large field mass is included. This may changesome of the details studied by Wang et al. [25], however, the result shouldnot be changed too much as the stress-energy tensor is still very similar.Using dimensional regularization drops the need for physical regularizationgiven that the meaningful parameter is now a field mass.Pauli-Villars in the context of stochastic gravity is largely ignored formany of the reasons discussed in this thesis. Dimensional regularization,on the other hand, is the default regularisation scheme in many studies ofstochastic gravity. Noise kernels for various systems of induced and intrinsicfluctuations have been calculated and could be applied in a case where thenoise kernel forms the basis for studying this problem. Using these tools, itmay be worth studying whether the dimensional regularization of a massivescalar field still gives the mathematical results of the parametric resonance.6.3 Future workIn this work, I have focused on questions about the use of stochastic gravityto study scattering and the cosmological constant problem. There are severalthings I have touched on that would be interesting for future work.\u2022 Rayleigh scattering - In Chapter 4, I used Rayleigh scattering asa close analogue to the sort of scattering one may expect in stochasticgravity. A very rough calculation assuming validity of this analogyshowed that the effect of scattering should be negligible. A more in-depth approach following from the same principles of Rayleigh scat-tering due to fluctuations could give definitive answers on the mangi-tude of scattering in stochastic gravity. Showing neglibile fluctuationswould mean that there is no experimental evidence from scattering forstochastic gravity, though this would make it consistent with observa-tions.\u2022 Parametric resonance in geodesic deviation - In Chapter 4, Idemonstrated the possibility for parametric resonance in studies oninterferometer experiments [69, 70]. Using both mode decomposition886.3. Future workand a stochastic gravity approach hint at the possibility of parametricresonance in these systems. Bak et al. [73] looked at higher-orderperturbations using noise kernel which was allowed to take negativevalues and got questionable results which a positive semi-definite noisekernel would not give. The demonstration of parametric resonance forgeodesics in stochastic gravity formalism could be a different approachto the one suggested by Wang et al. [25].\u2022 Induced metric fluctuations in interferometers - In Chapter4, I distinguished between the intrinsic metric perturbations used byParikh et al. [69] and Cho & Hu [70] and the induced metric perturba-tions assumed in Wang et al. [25]. Combining some of these ideas maygive a slightly different result to those discussed in either, and may beuseful for some applications such as studying parametric resonancefrom metric perturbations, though it is unlikely the detectability ofthese fluctuations in interferometer experiments would be much im-proved.\u2022 Amplification in Pauli-Villars - In Chapter 5, I discussed the pos-sibility of amplification due to the different magnitudes of coefficientsin Pauli-Villars. Such a system may allow a stable combination ofPauli-Villars fields which may allow some form of regularization usingthis procedure.\u2022 Alternative Pauli-Villars approaches - In Chapter 5, I brieflyoutlined an example of alternative approaches to formally definingPauli-Villars. To use Pauli-Villars in stochastic gravity, one shouldbe able to regularize expressions where at least two propagators aremultiplied. The approach outlined in Chapter 5 is not able to do thatfor massive fields, though there may be an approach that uses differenttools (such as matrices) to allow this regularization procedure andrespects the physical conditions.\u2022 Dimensional-regularization approach - In Chapter 6, I suggesteda different model for the source of fluctuations in the model used byWang et al. [25]. In Pauli-Villars, the highest-mass field is the sourceof the cutoff. Wang et al. used a regularized massless field, however, afield with a large mass could also gives similar results. In this case, aregularization method that does not give a cutoff - such as dimensionalregularization - could be used. Dimensional regularization has beenstudied in stochastic gravity much more than Pauli-Villars, and is well-896.3. Future workbehaved for calculating the noise kernel. 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Basic floquet theory.https:\/\/personal.math.ubc.ca\/ ward\/teaching\/m605\/every2f loquet1.pdf.97Appendix AMetric PerturbationCalculationsGiven a metric perturbation:g\u00b5\u03bd = \u03b7\u00b5\u03bd + h\u00b5\u03bd (A.1)One can write out expansions of various metric-dependent expressions:g\u00b5\u03bd = \u03b7\u00b5\u03bd +\u03b4g\u00b5\u03bd\u03b4g\u03b1\u03b2\u2223\u2223\u2223\u2223g=\u03b7h\u03b1\u03b2 +12\u03b42g\u00b5\u03bd\u03b4g\u03b1\u03b2\u03b4g\u03c3\u03c1\u2223\u2223\u2223\u2223g=\u03b7h\u03b1\u03b2h\u03c3\u03c1+16\u03b43g\u00b5\u03bd\u03b4g\u03b1\u03b2\u03b4g\u03c3\u03c1\u03b4g\u03b3\u03f5\u2223\u2223\u2223\u2223g=\u03b7h\u03b1\u03b2h\u03c3\u03c1h\u03b3\u03f5 +O(h4)= \u03b7\u00b5\u03bd + G\u00b5\u03bd\u03b1\u03b2h\u03b1\u03b2 + 12G\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1h\u03b1\u03b2h\u03c3\u03c1 + 16G\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5h\u03b1\u03b2h\u03c3\u03c1h\u03b3\u03f5 +O(h4)(A.2)\u221a\u2212g = 1 +\u221a\u2212g\u03b4g\u03b1\u03b2\u2223\u2223\u2223\u2223g=\u03b7h\u03b1\u03b2 +12\u03b42\u221a\u2212g\u03b4g\u03b1\u03b2\u03b4g\u03c3\u03c1\u2223\u2223\u2223\u2223g=\u03b7h\u03b1\u03b2h\u03c3\u03c1+16\u03b43\u221a\u2212g\u03b4g\u03b1\u03b2\u03b4g\u03c3\u03c1\u03b4g\u03b3\u03f5\u2223\u2223\u2223\u2223g=\u03b7h\u03b1\u03b2h\u03c3\u03c1h\u03b3\u03f5 +O(h4)= 1 + F\u03b1\u03b2h\u03b1\u03b2 + 12F\u03b1\u03b2\u03c3\u03c1h\u03b1\u03b2h\u03c3\u03c1 + 16F\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5h\u03b1\u03b2h\u03c3\u03c1h\u03b3\u03f5 +O(h4)(A.3)As a simplification (which is assumed throughout this work) one can usethe following:\u03b4g\u00b5\u03bd\u03b4g\u03b1\u03b2= \u221212(g\u00b5\u03b1g\u03bd\u03b2 + g\u00b5\u03b2g\u03bd\u03b1) = \u2212g\u00b5\u03b1g\u03bd\u03b2 (A.4)This simplification is justified so long as a trace is always taken over allbut one index pair, which is adequate in this work.98Appendix A. Metric Perturbation CalculationsCalculating the G tensors by taking variations with respect to the metricand setting g = \u03b7:G\u00b5\u03bd\u03b1\u03b2 = \u2212\u03b7\u00b5\u03b1\u03b7\u03bd\u03b2 , G\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1 = \u03b7\u00b5\u03c3\u03b7\u03c1\u03b1\u03b7\u03bd\u03b2 + \u03b7\u00b5\u03b1\u03b7\u03bd\u03c3\u03b7\u03c1\u03b2 (A.5)G\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5 = \u2212\u03b7\u00b5\u03b3\u03b7\u03f5\u03c3\u03b7\u03c1\u03b1\u03b7\u03bd\u03b2 \u2212 \u03b7\u00b5\u03c3\u03b7\u03c1\u03b3\u03b7\u03f5\u03b1\u03b7\u03bd\u03b2 \u2212 \u03b7\u00b5\u03c3\u03b7\u03c1\u03b1\u03b7\u03bd\u03b3\u03b7\u03f5\u03b2\u2212 \u03b7\u00b5\u03b3\u03b7\u03f5\u03b1\u03b7\u03bd\u03c3\u03b7\u03c1\u03b2 \u2212 \u03b7\u00b5\u03b1\u03b7\u03bd\u03b3\u03b7\u03f5\u03c3\u03b7\u03c1\u03b2 \u2212 \u03b7\u00b5\u03b1\u03b7\u03bd\u03c3\u03b7\u03c1\u03b3\u03b7\u03f5\u03b2 (A.6)G\u00b5\u03bd\u03b1\u03b2h\u03b1\u03b2 = \u2212h\u00b5\u03bd , 12G\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1h\u03b1\u03b2h\u03c3\u03c1 = h\u03c1\u03bdh\u00b5\u03c1 (A.7)16G\u00b5\u03bd\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5h\u03b1\u03b2h\u03c3\u03c1h\u03b3\u03f5 = \u2212h\u03c1\u03bdh\u03c3\u03c1h\u00b5\u03c3 (A.8)Following the same procedure to calculation the F tensors:F\u03b1\u03b2 =12\u03b7\u03b1\u03b2, F\u03b1\u03b2\u03c3\u03c1 =14\u03b7\u03b1\u03b2\u03b7\u03c3\u03c1 \u2212 12\u03b7\u03b1\u03c3\u03b7\u03b2\u03c1 (A.9)F\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5 = 18\u03b7\u03b3\u03f5\u03b7\u03b1\u03b2\u03b7\u03c3\u03c1 \u2212 14\u03b7\u03b3\u03c3\u03b7\u03f5\u03c1\u03b7\u03b1\u03b2 \u2212 14\u03b7\u03b3\u03f5\u03b7\u03b1\u03c3\u03b7\u03b2\u03c1\u2212 14\u03b7\u03c3\u03c1\u03b7\u03b3\u03b1\u03b7\u03f5\u03b2 +12\u03b7\u03b3\u03c3\u03b7\u03b1\u03c1\u03b7\u03f5\u03b2 +12\u03b7\u03b3\u03b1\u03b7\u03f5\u03c3\u03b7\u03b2\u03c1(A.10)There are some useful proofs one can demonstrate for the traces of thetensors. Taking Fg to be the tensor without taking the limit as g \u2192 \u03b7, onecan write:g\u03b1\u03b2F\u03b1\u03b2\u00b5\u03bd...g = g\u03b1\u03b2\u03b4(\u221a\u2212gF\u00b5\u03bd...g )\u03b4g\u03b1\u03b2= g\u03b1\u03b2(12\u221a\u2212gg\u03b1\u03b2F\u00b5\u03bd...g +\u221a\u2212g \u03b4F\u00b5\u03bd...g\u03b4g\u03b1\u03b2)(A.11)The tensor F\u00b5\u03bd...g contains terms proportional to various metric tensorswith raised indices such as g\u00b5\u03bd . By taking the trace of the variation, onewill get for every term:g\u03b1\u03b2\u03b4g\u00b5\u03bd\u03b4g\u03b1\u03b2= \u2212g\u03b1\u03b2g\u00b5\u03b1g\u03bd\u03b2 = \u2212g\u00b5\u03bd (A.12)Thus, in every term, only a minus sign is introduced. Finally, this canbe used to show:99Appendix A. Metric Perturbation Calculationsg\u03b1\u03b2F\u03b1\u03b2\u00b5\u03bd...g = 2\u221a\u2212gF\u00b5\u03bd...g +\u221a\u2212g \u03b4F\u00b5\u03bd...g\u03b4g\u03b1\u03b2\u2212\u221a\u2212gF\u00b5\u03bd...g =\u221a\u2212gF\u00b5\u03bd...g (A.13)This can be applied to any metric, so in the Minkowski metric one setsg = \u03b7 and\u221a\u2212g = 1, which simply eliminates the indices on the tensor. Thisis a simple case provided the pairs are respected - if one mixes indices fromdifferent pairs, the procedure is not as simple. Another exception is whenno indices remain after reducing - one just gets \u03b7\u00b5\u03bdF\u00b5\u03bd = 2, which can befound easily by substituting.One can use these tensors to find the expressions that are used in theexpansion when multiplied by the metric perturbations:F\u03b1\u03b2h\u03b1\u03b2 = 12h,12F\u03b1\u03b2\u03c3\u03c1h\u03b1\u03b2h\u03c3\u03c1 = 18h2 \u2212 14h\u03b1\u03b2h\u03b1\u03b2 (A.14)16F\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5h\u03b1\u03b2h\u03c3\u03c1h\u03b3\u03f5 = 148h3 \u2212 18hh\u03b1\u03b2h\u03b1\u03b2 +16h\u03b1\u03b2h\u03b1\u03c3h\u03c3\u03b2 (A.15)These results can therefore be used to write out some useful expansions,and the F tensor will show up frequently as it is a more condensed notationthat allows one to keep metric perturbations with lowered indices whichare defined precisely by (A.1). Substituting the expressions above into theformulas for g\u00b5\u03bd and\u221a\u2212g:g\u00b5\u03bd = \u03b7\u00b5\u03bd \u2212 h\u00b5\u03bd + h\u03c1\u00b5h\u03bd\u03c1 \u2212 h\u03c1\u03bdh\u03c3\u03c1h\u00b5\u03c3 +O(h4) (A.16)\u221a\u2212g = 1+ 12h+18(h2\u22122h\u03b1\u03b2h\u03b1\u03b2)+ 148(h3\u22126hh\u03b1\u03b2h\u03b1\u03b2+8h\u03b1\u03b2h\u03b1\u03c3h\u03c3\u03b2)+O(h4)(A.17)Going further by deriving some more useful expressions from these two:12\u221a\u2212gg\u03b1\u03b2 = \u03b4\u221a\u2212g\u03b4g\u03b1\u03b2= F\u03b1\u03b2 + F\u03b1\u03b2\u03c3\u03c1h\u03c3\u03c1 + 12F\u03b1\u03b2\u03c3\u03c1\u03b3\u03f5h\u03c3\u03c1h\u03b3\u03f5 +O(h3)=12\u03b7\u00b5\u03bd +14(\u03b7\u03b1\u03b2h\u2212 2h\u03b1\u03b2) + 116(\u03b7\u03b1\u03b2h2 \u2212 6hh\u03b1\u03b2 + 8h\u03b1\u03c3h\u03c3\u03b2) +O(h3)(A.18)100Appendix A. Metric Perturbation Calculations\u0393\u00b5\u03b1\u03b2 =12g\u00b5\u03bd(\u2202\u03b1g\u03bd\u03b2 + \u2202\u03b2g\u03bd\u03b1 \u2212 \u2202\u03bdg\u03b1\u03b2)=12(\u03b7\u00b5\u03bd \u2212 h\u00b5\u03bd + h\u03c1\u00b5h\u03bd\u03c1 \u2212 h\u03c1\u03bdh\u03c3\u03c1h\u00b5\u03c3)(\u2202\u03b1h\u03bd\u03b2 + \u2202\u03b2h\u03bd\u03b1 \u2212 \u2202\u03bdh\u03b1\u03b2) +O(h4)(A.19)All of this agrees with the calculations as demonstrated by Choi et al.[61], which has the Einstein-Hilbert action expanded up to quartic order inh in de Donder gauge. In this gauge \u2202\u00b5(\u221a\u2212gg\u00b5\u03bd) = 0, and one can defineh\u00b5\u03bd = h\u00b5\u03bd\u2212 12h\u03b7\u00b5\u03bd , which satisfies \u2202\u00b5h\u00b5\u03bd = 0. Using this coordinate change,the Ricci tensor and scalar and consequently the Einstein field equations canbe vastly simplified. To linear order, they are relatively straightforward (pg.300 in [41]):R\u00b5\u03bd = \u2202\u03bb\u0393\u03bb\u00b5\u03bd \u2212 \u2202\u03bd\u0393\u03bb\u00b5\u03bb + \u0393\u03bb\u00b5\u03bd\u0393\u03b3\u03bb\u03b3 \u2212 \u0393\u03bb\u00b5\u03b3\u0393\u03b3\u03bd\u03bb= \u2202\u03bb(12\u03b7\u03bb\u03c3(\u2202\u00b5h\u03c3\u03bd + \u2202\u03bdh\u03c3\u00b5 \u2212 \u2202\u03c3h\u00b5\u03bd))\u2212 \u2202\u03bd(12\u03b7\u03bb\u03c3(\u2202\u00b5h\u03c3\u03bb + \u2202\u03bbh\u03c3\u00b5 \u2212 \u2202\u03c3h\u00b5\u03bb))(A.20)R\u00b5\u03bd = \u221212\u25a1h\u00b5\u03bd , R = \u221212\u25a1h (A.21)R\u00b5\u03bd \u2212 12g\u00b5\u03bdR = \u221212\u25a1(h\u00b5\u03bd \u2212 12h\u03b7\u00b5\u03bd) + ... = \u221212\u25a1h\u00b5\u03bd + ... (A.22)The vacuum solutions can then be found by solving \u25a1h\u00b5\u03bd = 0, which isoften used for gravitational wave solutions. Looking at the Einstein-Hilbertaction to third order in h:SEH =\u222bd4x\u221a\u2212gR2\u03ba(A.23)SEH =1\u03ba\u222bd4x (\u221214\u2202\u00b5h\u2202\u00b5h+12\u2202\u00b5h\u03b1\u03b2\u2202\u00b5h\u03b1\u03b2)+ (12h\u03b1\u03b2\u2202\u00b5h\u03b2\u03b1\u2202\u00b5h\u221212h\u03b1\u03b2\u2202\u03b1h\u00b5\u03bd\u2202\u03b2h\u03bd\u00b5 \u2212 h\u03b1\u03b2\u2202\u00b5h\u03bd\u03b1\u2202\u00b5h\u03b2\u03bd+14h\u2202\u03b2h\u00b5\u03bd\u2202\u03b2h\u03bd\u00b5 + h\u03b2\u00b5\u2202\u03bdh\u03b1\u03b2\u2202\u00b5h\u03bd\u03b1 \u221218h\u2202\u03bdh\u2202\u03bdh) +O(h4)(A.24)101Appendix A. Metric Perturbation CalculationsThe first term can be interpreted as defining the graviton propagatorin Minkowski spacetime. Introducing the gauge removes the issues withdefining the propagator as it can be inverted. This term can first be writtenin terms of an operator:1\u03ba\u222bd4x (\u221214\u03b7\u00b5\u03bd\u03b7\u03b1\u03b2\u2202\u03bbh\u00b5\u03bd(x)\u2202\u03bbh\u03b1\u03b2(x)+14(\u03b7\u03b1\u00b5\u03b7\u03b2\u03bd + \u03b7\u03b1\u03bd\u03b7\u03b2\u00b5)\u2202\u03bbh\u00b5\u03bd(x)\u2202\u03bbh\u03b1\u03b2(x))=14\u03ba\u222bd4x d4(y)h\u00b5\u03bd(x)\u03b4(4)(x\u2212 y)(\u03b7\u00b5\u03bd\u03b7\u03b1\u03b2 \u2212 \u03b7\u03b1\u00b5\u03b7\u03b2\u03bd \u2212 \u03b7\u03b1\u03bd\u03b7\u03b2\u00b5)\u25a1h\u03b1\u03b2(y)=12\u03ba\u222bd4x d4y h\u00b5\u03bd(x)O\u00b5\u03bd\u03b1\u03b2(x, y)h\u03b1\u03b2(y)(A.25)Following the definition of the Green\u2019s function as shown in [62], theGreen\u2019s function can be found:O\u03b1\u03b2\u00b5\u03bdG\u03b1\u03b2\u03c3\u03c1(x\u2212 y) =14(\u03b7\u00b5\u03c3\u03b7\u03bd\u03c1 + \u03b7\u00b5\u03c1\u03b7\u03bd\u03c3)\u03b4(4)(x\u2212 y) (A.26)Defining P\u00b5\u03bd\u03b1\u03b2 =12(\u03b7\u03b1\u00b5\u03b7\u03b2\u03bd +\u03b7\u03b1\u03bd\u03b7\u03b2\u00b5\u2212\u03b7\u00b5\u03bd\u03b7\u03b1\u03b2), one can make an ansatzthat G is proportional to P to show this relationship holds:O\u03b1\u03b2\u00b5\u03bdG\u03b1\u03b2\u03c3\u03c1(x\u2212 y) = (1\u03ba\u03b7\u03b1\u03b3\u03b7\u03b2\u03f5P\u00b5\u03bd\u03b3\u03f5\u25a1)(\u03ba2P\u03b1\u03b2\u03c3\u03c1\u03b4(4)(x\u2212 y)(\u25a1)\u22121)=12(\u03b4\u03b1\u00b5\u03b4\u03b2\u03bd + \u03b4\u03b1\u03bd \u03b4\u03b2\u00b5 \u2212 \u03b4\u03b1\u03b2 \u03b4\u00b5\u03bd )(12P\u03b1\u03b2\u03c3\u03c1\u03b4(4)(x\u2212 y))=18(2\u03b7\u00b5\u03c3\u03b7\u03bd\u03c1 + 2\u03b7\u00b5\u03c1\u03b7\u03bd\u03c3)\u03b4(4)(x\u2212 y)) = 14(\u03b7\u00b5\u03c3\u03b7\u03bd\u03c1 + \u03b7\u00b5\u03c1\u03b7\u03bd\u03c3)\u03b4(4)(x\u2212 y)(A.27)G\u03b1\u03b2\u03c3\u03c1(x\u2212 y) = \u03ba2P\u03b1\u03b2\u03c3\u03c1\u03b4(4)(x\u2212 y))(\u25a1)\u22121, G\u02dc\u03b1\u03b2\u03c3\u03c1(k) = P\u03b1\u03b2\u03c3\u03c12k2(A.28)Looking at the third-order expression allows one to write out the Feyn-man rules for a three-graviton vertex. This is important to note as it demon-strates that the Einstein-Hilbert action contains self-interaction terms thatmean the theory is non-linear. This term can be used to generate loops forquantum corrections which have been studied to one-loop order [61, 62].102Appendix A. Metric Perturbation CalculationsIn the transverse-traceless gauge, extra conditions are included on topof the de Donder gauge. This includes setting h = 0 so that the quadraticterm in the action becomes:S(0)EH = \u221214\u03ba\u222bd4x \u2202\u00b5h\u03b1\u03b2\u2202\u00b5h\u03b1\u03b2 (A.29)This can be simplified further in a derivative gauge such as the transverse-traceless gauge.103Appendix BFloquet TheoryFloquet theory describes a system described by a periodic matrix P(t). Gen-erally, impulsive effects can be included, however, I will ignore those inthis analysis. The system of a parametric oscillator can be described by[72, 80, 81]:dxdt= P (t)x(t), P (t) =(0 1A cos(\u03c9t) 0), x(t) =(xx\u02d9)(B.1)This matrix of P (t) is periodic in T = 2\u03c0\u03c9 , and the system gives thefamiliar parametric oscillator equation as well as a trivial equation for x\u02d9:(x\u02d9x\u00a8)=(x\u02d9A cos(\u03c9t)x)(B.2)As P (t) is piecewise continuous and periodic, while the impulsive effectsare taken to be zero, this system meets the assumptions of Floquet theory[80]. The fundamental matrix of solutions is given as X(t) = {x1(t), x2(t)}.As P (t+ T ) = P (t), X(t+ T ) will also be a solution. It can be shown thatthere is a constant matrix B defined by X(t+ T ) = X(t)B so that:dX(t+ T )dt=ddt(X(t)B) = P (t)X(t)B (B.3)AsB is constant, one can define it by taking t=0, so thatB = X(0)\u22121X(T ).The determinant is shown to be [81]:det(B) = exp(\u222b T0tr(P (t))dt)(B.4)In this case, tr(P (t)) = 0, so det(B) = 1. By choosing a particularbasis for the solutions, one can show a relationship with the eigenvaluesof B, which are independent of the fundamental solution. First, settingthe boundary conditions of two solutions with x(0) = x0 and another withx\u02d9(0) = v0, then relating to the eigenvalues [72, 81]:104Appendix B. Floquet TheoryX(0) =(x0 00 v0), X(T ) =(x1(T ) x2(T )x\u02d91(T ) x\u02d92(T )), B =(x1(T )x0x2(T )x0x\u02d91(T )v0x\u02d92(T )v0)(B.5)It can be shown that the eigenvalues of B can be multipliers of particularsolutions, given by xn(t) = X(t)bn, where bn is an eigenvector of B witheigenvalue \u00b5n [81]:xn(t+ T ) = X(t+ T ) bn = X(t)Bbn = \u00b5nX(t)bn = \u00b5nxn(t) (B.6)To decompose this further, it can be shown there are periodic solutions\u03a0n(t + T ) = \u03a0n(t) so that xn(t) = e\u03bbntT \u03a0n(t) obeys this relationship for\u03bbn = ln(\u00b5n):xn(t+ T ) = e\u03bbntT e\u03bbn\u03a0n(t) = e\u03bbnx(t+ T ) = \u00b5nxn(t) (B.7)Given det(B) = 1, it must be that \u00b51\u00b52 = 1, so \u03bb2 = \u2212\u03bb1 = \u03bb = ln(\u00b5),picking |\u00b5| > 1. The general solution is then given by a linear combination:x(t) = \u00b5tT \u03a01(t) + \u00b5\u2212 tT \u03a02(t) (B.8)As the value of t grows larger, the first term will dominate, and so one willfind that solutions of this system have exponential growth over timescalesmuch larger than the period.105","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2023-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0435666","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Physics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/contributor":[{"value":"Unruh, W. 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