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1 Overview

Stochastic semiclassical gravity1 is a theory developed in the 1990s using semiclassical gravity (quantum fields in classical spacetimes, 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 gravity includes also its fluctuations in a new stochastic semiclassical or the Einstein–Langevin equation. 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 being 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 bi-tensor 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 on trans-Planckian physics, are new horizons to explore. On the 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 to explore the connection with high energy (Planckian) physics in the realm of quantum gravity.

In this review we summarize major work on 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 [148Jump To The Next Citation Point154Jump To The Next Citation Point]; a set of lectures which include a discussion of the issue of the validity of semiclassical gravity in [168Jump To The Next Citation Point]; a pedagogical introduction of stochastic gravity theory with a more complete treatment of backreaction problems in cosmology and black holes in [169Jump To The Next Citation Point]. A comprehensive formal description of the fundamentals is given in [207Jump To The Next Citation Point208Jump To The Next Citation Point] while that of the noise kernel in arbitrary spacetimes in [208Jump To The Next Citation Point243Jump To The Next Citation Point245Jump To The Next Citation Point]. Here we will try to mention all 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 where 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 [71Jump To The Next Citation Point200Jump To The Next Citation Point291Jump To The Next Citation Point] and techniques such as the influence functional [89Jump To The Next Citation Point88Jump To The Next Citation Point] (which is related to the closed-time-path effective action [257Jump To The Next Citation Point11Jump To The Next Citation Point184Jump To The Next Citation Point66Jump To The Next Citation Point272Jump To The Next Citation Point42Jump To The Next Citation Point70Jump To The Next Citation Point76Jump To The Next Citation Point181Jump To The Next Citation Point40Jump To The Next Citation Point182Jump To The Next Citation Point236Jump To The Next Citation Point]) were a great help in our understanding of the physical meaning of issues involved toward the construction of this new theory, foremost because quantum fluctuations and correlation have become the focus. Quantum statistical field theory and the statistical mechanics of quantum field theory [4143Jump To The Next Citation Point4547Jump 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 [215Jump To The Next Citation Point284Jump To The Next Citation Point] to semiclassical gravity.
    2. Quantum field theory in curved spacetimes [25Jump To The Next Citation Point100Jump To The Next Citation Point285Jump To The Next Citation Point113Jump To The Next Citation Point]:
      1. Stress-energy tensor: Regularization and renormalization.
      2. Self-consistent solution: Backreaction problems [203Jump To The Next Citation Point115Jump To The Next Citation Point158Jump To The Next Citation Point159Jump To The Next Citation Point124Jump To The Next Citation Point3Jump To The Next Citation Point4Jump To The Next Citation Point].
      3. Effective action: Closed time path, initial value formulation [257Jump To The Next Citation Point11Jump To The Next Citation Point184Jump To The Next Citation Point66Jump To The Next Citation Point272Jump To The Next Citation Point,  42Jump To The Next Citation Point70Jump To The Next Citation Point76Jump To The Next Citation Point181Jump To The Next Citation Point40Jump To The Next Citation Point182Jump To The Next Citation Point236Jump To The Next Citation Point].
      4. Equation of motion: Real and causal.
    3. Nonequilibrium statistical mechanics:
      1. Open quantum systems [71200291].
      2. Influence functional: Stochastic equations [89Jump To The Next Citation Point].
      3. Noise and decoherence: Quantum to classical transition [303Jump To The Next Citation Point304Jump To The Next Citation Point305Jump To The Next Citation Point306Jump To The Next Citation Point180Jump To The Next Citation Point,  33Jump To The Next Citation Point279Jump To The Next Citation Point307Jump To The Next Citation Point109Jump To The Next Citation Point114Jump To The Next Citation Point221Jump To The Next Citation Point222Jump To The Next Citation Point223Jump To The Next Citation Point224Jump To The Next Citation Point225Jump To The Next Citation Point226Jump To The Next Citation Point105Jump To The Next Citation Point125Jump To The Next Citation Point83Jump To The Next Citation Point120Jump To The Next Citation Point122Jump To The Next Citation Point,  30Jump To The Next Citation Point239Jump To The Next Citation Point278Jump To The Next Citation Point170Jump To The Next Citation Point171Jump To The Next Citation Point172Jump To The Next Citation Point121Jump To The Next Citation Point81Jump To The Next Citation Point82Jump To The Next Citation Point185Jump To The Next Citation Point186Jump To The Next Citation Point187Jump To The Next Citation Point173Jump To The Next Citation Point].
    4. Decoherence in quantum cosmology and emergence of classical spacetimes [188Jump To The Next Citation Point119Jump To The Next Citation Point228Jump To The Next Citation Point149Jump To The Next Citation Point36Jump To The Next Citation Point37Jump To The Next Citation Point160Jump To The Next Citation Point].
  2. Theory:
    1. Dissipation from particle creation [76Jump To The Next Citation Point181Jump To The Next Citation Point40Jump To The Next Citation Point182Jump To The Next Citation Point236Jump To The Next Citation Point57Jump To The Next Citation Point];
      backreaction as fluctuation-dissipation relation (FDR) [167Jump To The Next Citation Point].
    2. Noise from fluctuations of quantum fields [148Jump To The Next Citation Point150Jump To The Next Citation Point44Jump To The Next Citation Point].
    3. Einstein–Langevin equations [44Jump To The Next Citation Point157Jump To The Next Citation Point167Jump To The Next Citation Point58Jump To The Next Citation Point59Jump To The Next Citation Point38Jump To The Next Citation Point202Jump To The Next Citation Point207Jump To The Next Citation Point208Jump To The Next Citation Point206Jump To The Next Citation Point].
    4. Metric fluctuations in Minkowski spacetime [209Jump To The Next Citation Point].
  3. Issues:
    1. Validity of semiclassical gravity [163Jump To The Next Citation Point242Jump To The Next Citation Point].
    2. Viability of vacuum dominance and inflationary cosmology.
    3. Stress-energy bi-tensor and noise kernel: Regularization reassessed [243Jump To The Next Citation Point245Jump To The Next Citation Point].
  4. Applications: Early universe and black holes:
    1. Wave propagation in stochastic geometry [166Jump To The Next Citation Point].
    2. Black hole horizon fluctuations: Spontaneous/active versus induced/passive [94Jump To The Next Citation Point294Jump To The Next Citation Point,  266Jump To The Next Citation Point268Jump To The Next Citation Point14Jump To The Next Citation Point15Jump To The Next Citation Point211Jump To The Next Citation Point232Jump To The Next Citation Point245Jump To The Next Citation Point].
    3. Noise induced inflation [52].
    4. Structure formation [46Jump To The Next Citation Point213Jump To The Next Citation Point212Jump To The Next Citation Point39Jump To The Next Citation Point254Jump To The Next Citation Point];
      trace anomaly-driven inflation [269Jump To The Next Citation Point280Jump To The Next Citation Point132Jump To The Next Citation Point].
    5. Black hole backreaction as FDR [60Jump To The Next Citation Point258Jump To The Next Citation Point259Jump To The Next Citation Point217Jump To The Next Citation Point164Jump To The Next Citation Point54Jump To The Next Citation Point55Jump To The Next Citation Point264Jump To The Next Citation Point].
  5. Related Topics:
    1. Metric fluctuations and trans-Planckian problem [14Jump To The Next Citation Point15Jump To The Next Citation Point211232219].
    2. Spacetime fam [6263101102103].
    3. Universal ‘metric conductance’ fluctuations [261Jump To The Next Citation Point].
  6. Ideas:
    1. General relativity as geometro-hydrodynamics [152Jump To The Next Citation Point].
    2. Semiclassical gravity as mesoscopic physics [153].
    3. From stochastic to quantum gravity:
      1. Via correlation hierarchy of interacting quantum fields [154Jump To The Next Citation Point4347Jump To The Next Citation Point155Jump To The Next Citation Point].
      2. Possible relation to string theory and matrix theory.

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 the background material. We do not seek a complete coverage here, but will discuss only the selected topics in theory, issues, and applications. We use the (+, +, + ) sign conventions of [215284Jump To The Next Citation Point], and units in which c = ℏ = 1.


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