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 [148, 154]; a set of lectures which include a discussion of the issue of the validity of semiclassical gravity in [168]; a pedagogical introduction of stochastic gravity theory with a more complete treatment of backreaction problems in cosmology and black holes in [169]. A comprehensive formal description of the fundamentals is given in [207, 208] while that of the noise kernel in arbitrary spacetimes in [208, 243, 245]. 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 [71, 200, 291] and techniques such as the influence functional [89, 88] (which is related to the closed-time-path effective action [257, 11, 184, 66, 272, 42, 70, 76, 181, 40, 182, 236]) 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 [41, 43, 45, 47] 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 [215, 284] to semiclassical gravity.
   2. Quantum field theory in curved spacetimes [25, 100, 285, 113]:
      1. Stress-energy tensor: Regularization and renormalization.
      2. Self-consistent solution: Backreaction problems [203, 115, 158, 159, 124, 3, 4].
      3. Effective action: Closed time path, initial value formulation [257, 11, 184, 66, 272,  42, 70, 76, 181, 40, 182, 236].
      4. Equation of motion: Real and causal.
   3. Nonequilibrium statistical mechanics:
      1. Open quantum systems [71, 200, 291].
      2. Influence functional: Stochastic equations [89].
      3. Noise and decoherence: Quantum to classical transition [303, 304, 305, 306, 180,  33, 279, 307, 109, 114, 221, 222, 223, 224, 225, 226, 105, 125, 83, 120, 122,  30, 239, 278, 170, 171, 172, 121, 81, 82, 185, 186, 187, 173].
   4. Decoherence in quantum cosmology and emergence of classical spacetimes [188, 119, 228, 149, 36, 37, 160].
2. Theory:
   1. Dissipation from particle creation [76, 181, 40, 182, 236, 57];
      backreaction as fluctuation-dissipation relation (FDR) [167].
   2. Noise from fluctuations of quantum fields [148, 150, 44].
   3. Einstein–Langevin equations [44, 157, 167, 58, 59, 38, 202, 207, 208, 206].
   4. Metric fluctuations in Minkowski spacetime [209].
3. Issues:
   1. Validity of semiclassical gravity [163, 242].
   2. Viability of vacuum dominance and inflationary cosmology.
   3. Stress-energy bi-tensor and noise kernel: Regularization reassessed [243, 245].
4. Applications: Early universe and black holes:
   1. Wave propagation in stochastic geometry [166].
   2. Black hole horizon fluctuations: Spontaneous/active versus induced/passive [94, 294,  266, 268, 14, 15, 211, 232, 245].
   3. Noise induced inflation [52].
   4. Structure formation [46, 213, 212, 39, 254];
      trace anomaly-driven inflation [269, 280, 132].
   5. Black hole backreaction as FDR [60, 258, 259, 217, 164, 54, 55, 264].
5. Related Topics:
   1. Metric fluctuations and trans-Planckian problem [14, 15, 211, 232, 219].
   2. Spacetime fam [62, 63, 101, 102, 103].
   3. Universal ‘metric conductance’ fluctuations [261].
6. Ideas:
   1. General relativity as geometro-hydrodynamics [152].
   2. Semiclassical gravity as mesoscopic physics [153].
   3. From stochastic to quantum gravity:
      1. Via correlation hierarchy of interacting quantum fields [154, 43, 47, 155].
      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 [215, 284], and units in which .