

1 Overview
Stochastic semiclassical gravity 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 [149
, 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
, 244
, 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
, 41
, 70
, 76
, 181
, 39
, 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 [40, 42
, 44, 46
] 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:
- Ingredients:
- From general relativity [215
, 285
] to semiclassical
gravity.
- Quantum field theory in curved
spacetimes [25
, 100
, 286
, 113
]:
- Stress-energy tensor: Regularization and
renormalization.
- Self-consistent solution: Backreaction
problems [203
, 115
, 158
, 159
, 124
, 3
, 4
].
- Effective action: Closed time path, initial
value formulation [257
, 11
, 184
, 66
, 272
, 41
, 70
, 76
, 181
, 39
, 182
, 236
].
- Equation of motion: Real and causal.
- Nonequilibrium statistical mechanics:
- Open quantum systems [71, 200, 291].
- Influence functional: Stochastic
equations [89
].
- 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
].
- Decoherence in quantum cosmology and emergence
of classical spacetimes [188
, 119
, 228
, 150
, 36
, 37
, 160
].
- Theory:
- Dissipation from particle creation [76
, 181
, 39
, 182
, 236
, 57
];
backreaction as fluctuation-dissipation relation (FDR) [167
].
- Noise from fluctuations of quantum
fields [149
, 151
, 43
].
- Einstein-Langevin equations [43
, 157
, 167
, 58
, 59
, 38
, 202
, 207
, 208
, 206
].
- Metric fluctuations in Minkowski
spacetime [209
].
- Issues:
- Validity of semiclassical gravity [163
, 243
].
- Viability of vacuum dominance and inflationary
cosmology.
- Stress-energy bi-tensor and noise kernel:
Regularization reassessed [244
, 245
].
- Applications: Early
universe and black holes:
- Wave propagation in stochastic
geometry [166
].
- Black hole horizon fluctuations:
Spontaneous/active versus induced/passive [94
, 294
, 267
, 268
, 14
, 15
, 211
, 232
, 245
].
- Noise induced inflation [50].
- Structure formation [45
, 213
, 212
, 51
, 254
];
trace anomaly-driven inflation [269
, 280
, 132
].
- Black hole backreaction as FDR [60
, 258
, 259
, 217
, 164
, 54
, 55
, 264
].
- Related Topics:
- Metric fluctuations and trans-Planckian
problem [14
, 15
, 211, 232, 219].
- Spacetime fam [62, 63, 101, 102, 103].
- Universal ‘metric conductance’
fluctuations [261
].
- Ideas:
- General relativity as
geometro-hydrodynamics [146
].
- Semiclassical gravity as mesoscopic
physics [153].
- From stochastic to quantum gravity:
- Via correlation hierarchy of interacting
quantum fields [154
, 42, 46
, 155
].
- 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, 285
], and units in which
.

