The second step in the description of the
interaction of gravity with quantum fields is backreaction, i.e.,
the effect of the quantum fields on the spacetime geometry. The
source here is the expectation value of the stress-energy operator
for the 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 [75, 67
, 68
]. 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 [283
, 284
] 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 on the
spacetime is thus the central task in semiclassical gravity. One
assumes a general class of spacetime where the quantum fields live
in and act on, and seek a solution which 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
the source is known as the semiclassical
Einstein equation. Semiclassical gravity was first
investigated in cosmological backreaction problems [203, 115
, 158
, 159
, 124
, 3
, 4
, 123
, 90
, 129
]; an example is the
damping of anisotropy in Bianchi universes by the backreaction of
vacuum particle creation. Using the effect of quantum field
processes such as particle creation to explain why the universe is
so isotropic at the present was investigated in the context of
chaotic cosmology [214, 19, 20] in the
late 1970s prior to the inflationary cosmology proposal of the
1980s [117, 2, 197, 198], which assumes the
vacuum expectation value of an inflaton field as the source,
another, perhaps more well-known, example of semiclassical gravity.