Go to previous page Go up Go to next page

1 Overview

Stochastic semiclassical gravity is a theory developed in the 1990s using semiclassical gravity (quantum fields in classical spacetimes, the dynamics of both matter and spacetime are solved self-consistently) as the starting point and aiming at a theory of quantum gravity as the goal. While semiclassical gravity is based on the semiclassical Einstein equation with the source given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity, or stochastic gravity for short, also includes its fluctuations in a new stochastic semiclassical Einstein–Langevin equation (we will often use the shortened term stochastic gravity as there is no confusion as to the nature and source of stochasticity in gravity being induced from the quantum fields and not a priori from the classical spacetime). If the centerpiece in semiclassical-gravity theory is the vacuum expectation value of the stress-energy tensor of a quantum field and the central issues are how well the vacuum is defined and how the divergences can be controlled by regularization and renormalization, the centerpiece in stochastic semiclassical-gravity theory is the stress-energy bitensor and its expectation value known as the noise kernel. The mathematical properties of this quantity and its physical content in relation to the behavior of fluctuations of quantum fields in curved spacetimes are the central issues of this new theory. How they induce metric fluctuations and seed the structures of the universe, how they affect the black-hole horizons and the backreaction of Hawking radiance in black hole dynamics, including implications for trans-Planckian physics, are new horizons to explore. On theoretical issues, stochastic gravity is the necessary foundation to investigate the validity of semiclassical gravity and the viability of inflationary cosmology based on the appearance and sustenance of a vacuum energy-dominated phase. It is also a useful beachhead supported by well-established low-energy (sub-Planckian) physics from which to explore the connection with high-energy (Planckian) physics in the realm of quantum gravity.

In this review we summarize the major work and results of this theory since 1998. It is in the nature of a progress report rather than a review. In fact we will have room only to discuss a handful of topics of basic importance. A review of ideas leading to stochastic gravity and further developments originating from it can be found in [181Jump To The Next Citation Point, 187Jump To The Next Citation Point], a set of lectures, which include a discussion of the issue of the validity of semiclassical gravity in [207Jump To The Next Citation Point] and a pedagogical introduction of stochastic-gravity theory with a more detailed treatment of backreaction problems in cosmology and black holes in quasi-equilibrium in [208Jump To The Next Citation Point]. A comprehensive formal description of the fundamentals is given in [257Jump To The Next Citation Point, 258Jump To The Next Citation Point], while that of the noise kernel in arbitrary spacetimes can be found in [258Jump To The Next Citation Point, 304Jump To The Next Citation Point, 305Jump To The Next Citation Point]. Here we will try to mention related work so the reader can at least trace out the parallel and sequential developments. The references at the end of each topic below are representative work in which one can seek out further treatments.

Stochastic gravity theory is built on three pillars: general relativity, quantum fields and nonequilibrium statistical mechanics. The first two uphold semiclassical gravity, the last two span statistical field theory. Strictly speaking one can understand a great deal without appealing to statistical mechanics, and we will try to do so here. But concepts such as quantum open systems [88Jump To The Next Citation Point, 246Jump To The Next Citation Point, 370Jump To The Next Citation Point] and techniques such as the influence functional [107Jump To The Next Citation Point, 108Jump To The Next Citation Point] (which is related to the closed-time-path effective action [14Jump To The Next Citation Point, 54Jump To The Next Citation Point, 56Jump To The Next Citation Point, 82Jump To The Next Citation Point, 87Jump To The Next Citation Point, 94Jump To The Next Citation Point, 222Jump To The Next Citation Point, 223Jump To The Next Citation Point, 227Jump To The Next Citation Point, 296Jump To The Next Citation Point, 323Jump To The Next Citation Point, 343Jump To The Next Citation Point]) were a great help in our understanding of the physical meaning of issues involved in the construction of this new theory. Foremost because quantum fluctuations and correlation have ascended the stage and become the focus of attention. Quantum statistical field theory and the statistical mechanics of quantum field theory [55, 57Jump To The Next Citation Point, 59, 61Jump To The Next Citation Point] also aided us in searching for the connection with quantum gravity through the retrieval of correlations and coherence.

We show the scope of stochastic gravity as follows:

  1. Ingredients:
    1. From General Relativity [266Jump To The Next Citation Point, 361Jump To The Next Citation Point] to Semiclassical Gravity.
    2. Quantum Field Theory in Curved Spacetimes [34Jump To The Next Citation Point, 121Jump To The Next Citation Point, 135Jump To The Next Citation Point, 362Jump To The Next Citation Point]:
      1. Stress-energy tensor: Regularization and renormalization.
      2. Self-consistent solution: Backreaction problems in early universe and black holes [3Jump To The Next Citation Point, 4Jump To The Next Citation Point, 5, 109Jump To The Next Citation Point, 137Jump To The Next Citation Point, 147Jump To The Next Citation Point, 148Jump To The Next Citation Point, 153Jump To The Next Citation Point, 154Jump To The Next Citation Point, 165Jump To The Next Citation Point, 193Jump To The Next Citation Point, 194Jump To The Next Citation Point, 251Jump To The Next Citation Point], and analog gravity [15Jump To The Next Citation Point, 16Jump To The Next Citation Point, 252, 320Jump To The Next Citation Point, 321].
      3. Effective action: Closed time path, initial value formulation [14Jump To The Next Citation Point, 54Jump To The Next Citation Point, 56Jump To The Next Citation Point, 82Jump To The Next Citation Point, 87Jump To The Next Citation Point, 94Jump To The Next Citation Point, 223Jump To The Next Citation Point, 227Jump To The Next Citation Point, 296Jump To The Next Citation Point, 323Jump To The Next Citation Point, 343Jump To The Next Citation Point].
      4. Equation of motion: Real and causal [222Jump To The Next Citation Point].
    3. Nonequilibrium Statistical Mechanics (see [62] and references therein) :
      1. Open quantum systems [88, 246, 370].
      2. Influence Functional: Stochastic equations [107Jump To The Next Citation Point, 108Jump To The Next Citation Point].
      3. Noise and Decoherence: Quantum to classical transition [43Jump To The Next Citation Point, 46Jump To The Next Citation Point, 99Jump To The Next Citation Point, 100Jump To The Next Citation Point, 101Jump To The Next Citation Point, 126Jump To The Next Citation Point, 131Jump To The Next Citation Point, 136Jump To The Next Citation Point, 144Jump To The Next Citation Point, 145Jump To The Next Citation Point, 146Jump To The Next Citation Point, 149Jump To The Next Citation Point, 209Jump To The Next Citation Point, 210Jump To The Next Citation Point, 211Jump To The Next Citation Point, 212Jump To The Next Citation Point, 221Jump To The Next Citation Point, 228Jump To The Next Citation Point, 229Jump To The Next Citation Point, 230Jump To The Next Citation Point, 275Jump To The Next Citation Point, 276Jump To The Next Citation Point, 277Jump To The Next Citation Point, 278Jump To The Next Citation Point, 279Jump To The Next Citation Point, 280Jump To The Next Citation Point, 299Jump To The Next Citation Point, 350Jump To The Next Citation Point, 352Jump To The Next Citation Point, 389Jump To The Next Citation Point, 390Jump To The Next Citation Point, 391Jump To The Next Citation Point, 392Jump To The Next Citation Point, 393Jump To The Next Citation Point].
    4. Decoherence in Quantum Cosmology and Emergence of Classical Spacetimes [50Jump To The Next Citation Point, 51Jump To The Next Citation Point, 143Jump To The Next Citation Point, 182Jump To The Next Citation Point, 195Jump To The Next Citation Point, 231Jump To The Next Citation Point, 283Jump To The Next Citation Point].
  2. Theory:
    1. Dissipation from Particle Creation [54Jump To The Next Citation Point, 72Jump To The Next Citation Point, 94Jump To The Next Citation Point, 222Jump To The Next Citation Point, 223Jump To The Next Citation Point, 296Jump To The Next Citation Point];
      Backreaction as Fluctuation-Dissipation Relation (FDR) [69Jump To The Next Citation Point, 76Jump To The Next Citation Point, 206Jump To The Next Citation Point, 268Jump To The Next Citation Point].
    2. Noise from Fluctuations of Quantum Fields [58Jump To The Next Citation Point, 181Jump To The Next Citation Point, 183Jump To The Next Citation Point].
    3. Einstein–Langevin Equations [52Jump To The Next Citation Point, 58Jump To The Next Citation Point, 73Jump To The Next Citation Point, 74Jump To The Next Citation Point, 192Jump To The Next Citation Point, 206Jump To The Next Citation Point, 248Jump To The Next Citation Point, 256Jump To The Next Citation Point, 257Jump To The Next Citation Point, 258Jump To The Next Citation Point].
    4. Metric Fluctuations in Minkowski spacetime [259Jump To The Next Citation Point].
  3. Issues:
    1. Validity of Semiclassical Gravity [10Jump To The Next Citation Point, 111Jump To The Next Citation Point, 169Jump To The Next Citation Point, 170Jump To The Next Citation Point, 198Jump To The Next Citation Point, 203Jump To The Next Citation Point, 204Jump To The Next Citation Point, 223Jump To The Next Citation Point, 237Jump To The Next Citation Point, 303Jump To The Next Citation Point, 329Jump To The Next Citation Point].
    2. Viability of Vacuum Dominance and Inflationary Cosmology.
    3. Stress-Energy Bitensor and Noise Kernel: Regularization Reassessed [304Jump To The Next Citation Point, 305Jump To The Next Citation Point].
  4. Applications: Early Universe and Black Holes:
    1. Wave Propagation in Stochastic Geometry [205Jump To The Next Citation Point].
    2. Black Hole Horizon Fluctuations: Spontaneous/Active versus Induced/Passive [20Jump To The Next Citation Point, 21Jump To The Next Citation Point, 114Jump To The Next Citation Point, 261Jump To The Next Citation Point, 289Jump To The Next Citation Point, 305Jump To The Next Citation Point, 336Jump To The Next Citation Point, 338Jump To The Next Citation Point, 377Jump To The Next Citation Point].
    3. Noise induced inflation [67].
    4. Structure Formation [53Jump To The Next Citation Point, 60Jump To The Next Citation Point, 262Jump To The Next Citation Point, 263Jump To The Next Citation Point, 316Jump To The Next Citation Point];
      trace anomaly-driven inflation [162Jump To The Next Citation Point, 339Jump To The Next Citation Point, 355Jump To The Next Citation Point].
    5. Black Hole Backreaction and Fluctuations [69Jump To The Next Citation Point, 70Jump To The Next Citation Point, 76Jump To The Next Citation Point, 199Jump To The Next Citation Point, 200Jump To The Next Citation Point, 201Jump To The Next Citation Point, 268Jump To The Next Citation Point, 324Jump To The Next Citation Point, 325Jump To The Next Citation Point, 332Jump To The Next Citation Point].
  5. Related Topics:
    1. Metric Fluctuations and Trans-Planckian Problem [20Jump To The Next Citation Point, 21Jump To The Next Citation Point, 261, 273, 289Jump To The Next Citation Point].
    2. Spacetime Foam, Loop and Spin Foam [32, 78, 79, 85, 116, 122, 123, 124, 271, 272].
    3. Universal ‘Metric Conductance’ Fluctuations  [328Jump To The Next Citation Point].
  6. Ideas:
    1. General Relativity as Geometro-Hydrodynamics [105Jump To The Next Citation Point, 164Jump To The Next Citation Point, 185Jump To The Next Citation Point, 189, 216Jump To The Next Citation Point, 356Jump To The Next Citation Point, 357Jump To The Next Citation Point];
      Emergent Gravity [139Jump To The Next Citation Point, 173Jump To The Next Citation Point, 240Jump To The Next Citation Point, 326Jump To The Next Citation Point].
    2. Semiclassical Gravity as Mesoscopic Physics [186Jump To The Next Citation Point, 190Jump To The Next Citation Point].
    3. From Stochastic to Quantum Gravity:
      1. Via Correlation hierarchy of interacting quantum fields [57, 61Jump To The Next Citation Point, 187Jump To The Next Citation Point, 188Jump To The Next Citation Point].
      2. Possible relation to string theory and matrix theory.
      3. Other major approaches to quantum gravity [281].

For lack of space we list only the latest work in the respective topics above, describing ongoing research. The reader should consult the references therein for earlier work and background material. We do not seek a complete coverage here, but will discuss only those selected topics in theory, issues and applications. We use the (+, +, + ) sign conventions of [266, 361Jump To The Next Citation Point], and units in which c = ℏ = 1.


  Go to previous page Go up Go to next page