Semiclassical and Stochastic Gravity: Quantum Field Effects on Curved Spacetime

Description

The two pillars of modern physics are general relativity and quantum field theory, the former describes the large scale structure and dynamics of space-time, the latter, the microscopic constituents of matter. Combining the two yields quantum field theory in curved space-time, which is needed to understand quantum field processes in the early universe and black holes, such as the well-known Hawking effect. This book examines the effects of quantum field processes back-reacting on the background space-time which become important near the Planck time (10-43 sec). It explores the self-consistent description of both space-time and matter via the semiclassical Einstein equation of semiclassical gravity theory, exemplified by the inflationary cosmology, and fluctuations of quantum fields which underpin stochastic gravity, necessary for the description of metric fluctuations (space-time foams). Covering over four decades of thematic development, this book is a valuable resource for researchers interested in quantum field theory, gravitation and cosmology. Contents......Page 9 Preface......Page 13 1 Overview: Main Themes. Key Issues. Reader’s Guide......Page 16 1.1 From QFT in Curved Spacetime to Semiclassical and Stochastic Gravity......Page 17 1.2 Quantum, Stochastic, Semiclassical......Page 30 1.3 Low Energy Limit: Relation to Gravitational Quantum Physics......Page 35 1.4 Emphasis and Approach. Guide to the Reader......Page 43 Part I Effective Action and Regularization, Stress Tensor and Fluctuations......Page 50 2 ‘In-Out’ Effective Action. Dimensional Regularization......Page 52 2.1 Quantum Field Theory in Dynamical Spacetimes: Key Points......Page 53 2.2 The Schwinger–DeWitt (‘In-Out’) Effective Action......Page 61 2.3 Effective Action of an Interacting Field: Particle Creation and Interaction......Page 71 2.4 Quasilocal Effective Action for Slowly Varying Background......Page 79 2.5 Regularization of Quasilocal Lagrangian for λΦ4 Field......Page 86 2.6 Renormalization Group Equations......Page 90 3.1 The ‘In-In’ Effective Action......Page 94 3.2 ‘In-In’ Formalism for Quantum Fields in Curved Spacetime......Page 102 3.3 In-In Effective Action in Bianchi Type I Universe......Page 104 3.4 Expectation Value of the Stress Energy Tensor for Interacting Fields......Page 116 3.5 CTP Effective Action for Thermal Fields......Page 121 4 Stress-Energy Tensor and Correlators: Zeta-Function Method......Page 128 4.1 Zeta Function Regularization of 1-loop Effective Potential......Page 129 4.2 One-Loop Finite Temperature Effective Potential......Page 133 4.3 One-Loop Effective Potential in the Einstein Universe......Page 136 4.4 O(N) Self-Interacting Scalar Field in Curved Spacetime......Page 141 4.5 Stress-Energy 2-Pt Function from 2nd Variation of Effective Action......Page 148 4.6 Energy Density Fluctuations in Σ = Rd ×S1......Page 155 4.7 Correlations of the Stress-Energy Tensor in AdS Space......Page 159 5 Stress-Energy Tensor and Correlation. Point Separation......Page 165 5.1 Stress-Energy Bitensors and Products......Page 166 5.2 Point-Separation Regularization of the Stress-Energy Tensor......Page 175 5.3 The Noise Kernel: Structure, Forms and Computations......Page 184 Part II Infrared Behavior, 2PI, I/N, Backreaction and Semiclassical Gravity......Page 198 6.1 Overview: Relevance, Issues and Approaches......Page 200 6.2 Euclidean Zero-Mode, EIRD, 2PI Effective Action......Page 211 6.3 Lorentzian de Sitter: Late Time IR and Stochastic Approach......Page 222 6.4 Nonperturbative RG. Graviton and Gauge Issues......Page 236 7 Advanced Field Theory Topics......Page 243 7.1 2PI CTP Effective Action in Curved Spacetime......Page 244 7.2 The O(N) Field Theory in Curved Spacetime......Page 255 7.3 Remark: Consistent Renormalization of 2PI Effective Action......Page 267 7.4 Solving the Gap Equation for the Infrared Behavior of O(N) Field in dS......Page 268 7.5 Yukawa Coupled Scalar and Spinor Fields in Curved Spacetime......Page 270 8 Backreaction of Early Universe Quantum Processes......Page 280 8.1 Vacuum Energy-Driven Cosmology......Page 281 8.2 Backreaction of Cosmological Particle Creation......Page 298 8.3 Preheating from Inflaton Particle Creation and Interaction......Page 310 8.4 Other Examples: Stochastic Inflation, Minisuperspace Cosmology......Page 326 Part III Stochastic Gravity......Page 330 9 Metric Correlations at One-Loop: In-In and Large N......Page 332 9.1 The In-In Formalism in Flat Spacetime......Page 333 9.2 The In-In or CTP Effective Action......Page 339 9.3 Gravity and Matter Interaction in the CTP Formalism......Page 342 9.4 Large N Expansion: A Toy Model......Page 349 10 The Einstein–Langevin Equation......Page 352 10.1 Semiclassical Gravity: Axiomatic Approach......Page 353 10.2 Stochastic Gravity: Axiomatic Approach......Page 357 10.3 Validity of Semiclassical Gravity......Page 364 10.4 Functional Approach......Page 369 10.5 Explicit Form of the Einstein–Langevin Equation......Page 375 11 Metric Fluctuations in Minkowski Spacetime......Page 379 11.1 Perturbations around Minkowski Spacetime......Page 380 11.2 The Kernels in the Minkowski Background......Page 382 11.3 Einstein–Langevin Equation......Page 386 11.4 Solutions of the Einstein–Langevin Equation......Page 389 11.5 Stability of Minkowski Spacetime......Page 396 Part IV Cosmological and Black Hole Backreaction with Fluctuations......Page 404 12 Cosmological Backreaction with Fluctuations......Page 406 12.1 The Backreaction Problem in Cosmology......Page 407 12.2 Influence Action for Cosmological Perturbations......Page 409 12.3 Einstein–Langevin Equation......Page 413 12.4 Detailed Computation of the Trace Terms......Page 416 12.5 Mathematical Supplement......Page 420 13 Structure Formation in the Early Universe......Page 425 13.1 The Model......Page 426 13.2 Einstein–Langevin Equation for Scalar Perturbations......Page 427 13.3 Correlation for Scalar Perturbations......Page 431 13.4 Equivalence of the Stochastic–Quantum Correlations......Page 433 13.5 Including One-Loop Contributions......Page 436 14.1 Issues, Proposals and Scenarios......Page 438 14.2 General Issues on Backreaction......Page 449 14.3 Backreaction under Quasi-Static Conditions......Page 454 14.4 Metric Fluctuations of an Evaporating Black Hole......Page 467 14.5 Work on Metric Fluctuations without Backreaction......Page 477 Part V Quantum Curvature Fluctuations in de Sitter Spacetime......Page 480 15 Stress-Energy Tensor Fluctuations in de Sitter Space......Page 482 15.1 De Sitter Geometry and Invariant Bitensors......Page 483 15.2 Noise Kernel in de Sitter Spacetime......Page 488 15.3 Analysis Based on Field Modes......Page 492 15.4 Implications for Gravitational Fluctuations......Page 495 16 Two-Point Metric Perturbations in de Sitter......Page 498 16.1 Effective Action for Cosmological Perturbations......Page 499 16.2 Classification of Metric Perturbations......Page 504 16.3 Two-Point Functions for Tensor Metric Perturbations......Page 509 16.4 Intrinsic/Induced Fluctuations and Secular Terms......Page 516 16.5 Two-Point Functions for Metric Perturbations......Page 520 16.6 Effective Action for a General Conformal Field Theory......Page 524 16.7 Mathematical Supplement......Page 528 17 Riemann Tensor Correlator in de Sitter......Page 534 17.1 De Sitter-Invariant Bitensors......Page 535 17.2 Correlators of Curvature Tensors......Page 537 17.3 Riemann Tensor Correlator for General CFTs......Page 548 17.4 Riemann Tensor Correlator in Minkowski Spacetime......Page 551 17.5 Useful Fourier Transforms......Page 553 18.1 A New Perspective and Two Different Routes......Page 555 18.2 Emergent versus Quantum Gravity......Page 557 18.3 Unraveling the Microstructures of Spacetime......Page 559 18.4 Relation to Quantum Gravity and Limitations of Stochastic Gravity......Page 562 References......Page 565 Index......Page 606