There are three main steps that led to the recent development of stochastic gravity. The first step begins with quantum field theory in curved spacetime [34, 93, 121, 135, 362], which describes the behavior of quantum matter fields propagating in a specified (not dynamically determined by the quantum matter field as source) background gravitational field. In this theory the gravitational field is given by the classical spacetime metric determined from classical sources by the classical Einstein equations and the quantum fields propagate as test fields in such a spacetime. Some important processes described by quantum field theory in curved spacetime are particle creation from the vacuum, effects of vacuum fluctuations and polarizations in the early universe [29, 30, 31, 81, 93, 117, 179, 293, 327, 386, 387], and Hawking radiation in black holes [155, 156, 213, 294, 358].
The second step in the description of the interaction of gravity with quantum fields is backreaction, i.e., the effect of quantum fields on spacetime geometry. The source here is the expectation value of the stress-energy operator for matter fields in some quantum state in the spacetime, a classical observable. However, since this object is quadratic in the field operators, which are only well-defined as distributions on the spacetime, it involves ill-defined quantities. It contains ultraviolet divergences, the removal of which requires a renormalization procedure [83, 84, 93]. The final expectation value of the stress-energy operator using a reasonable regularization technique is essentially unique, modulo some terms, which depend on the spacetime curvature and which are independent of the quantum state. This uniqueness was proved by Wald [359, 360] who investigated the criteria that a physically meaningful expectation value of the stress-energy tensor ought to satisfy.
The theory obtained from a self-consistent solution of the geometry of the spacetime and the quantum field is known as semiclassical gravity. Incorporating the backreaction of the quantum matter field into the spacetime is thus the central task in semiclassical gravity. One assumes a general class of spacetime, in which the quantum fields live and on which they act and seek a solution that satisfies simultaneously the Einstein equation for the spacetime and the field equations for the quantum fields. The Einstein equation, which has the expectation value of the stress-energy operator of the quantum matter field as its source, is known as the semiclassical Einstein equation. Semiclassical gravity was first investigated in cosmological backreaction problems [3, 4, 109, 137, 147, 148, 153, 154, 193, 194, 251]; an example is the damping of anisotropy in Bianchi universes by the backreaction of vacuum particle creation. The effect of quantum field processes, such as particle creation, was used to explain why the universe is so isotropic at the present in the context of chaotic cosmology [27, 28, 265] in the late 1970s prior to the inflationary-cosmology proposal of the 1980s [2, 140, 243, 244], which assumes the vacuum expectation value of an inflaton field as the source, another, perhaps more well-known, example of semiclassical gravity.
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