Once the fluctuations of the stress-energy
operator have been characterized, we can perturbatively extend the
semiclassical theory to account for such fluctuations. Thus we will
assume that the background spacetime metric is a solution of the semiclassical Einstein
Equations (7
), and we will write
the new metric for the extended theory as
, where we
will assume that
is a perturbation to the background
solution. The renormalized stress-energy operator and the state of
the quantum field may now be denoted by
and
, respectively, and
will be the corresponding expectation value.
Let us now introduce a Gaussian stochastic tensor
field defined by the following correlators:
An important property of this stochastic tensor
is that it is covariantly conserved in the background spacetime,
. In fact, as a consequence of the
conservation of
one can see that
. Taking the divergence in
Equation (13
) one can then show
that
and
, so that
is deterministic and represents with certainty the
zero vector field in
.
For a conformal field, i.e., a field whose
classical action is conformally invariant, is traceless:
;
thus, for a conformal matter field the stochastic source gives no
correction to the trace anomaly. In fact, from the trace anomaly
result which states that
is, in
this case, a local c-number functional of
times the identity operator, we have that
. It then follows from Equation (13
) that
and
; an alternative proof based on the point-separation
method is given in [244
, 245
] (see also
Section 5).
All these properties make it quite natural to
incorporate into the Einstein equations the stress-energy
fluctuations by using the stochastic tensor as the source of the metric perturbations. Thus we
will write the following equation:
Note that we refer to the Einstein-Langevin
equation as a first order extension to the semiclassical Einstein
equation of semiclassical gravity and the lowest level
representation of stochastic gravity. However, stochastic gravity
has a much broader meaning, as it refers to the range of theories
based on second and higher order correlation functions. Noise can
be defined in effectively open systems (e.g., correlation
noise [46] in the
Schwinger-Dyson equation hierarchy) to some degree, but one should
not expect the Langevin form to prevail. In this sense we say that
stochastic gravity is the intermediate theory between semiclassical
gravity (a mean field theory based on the expectation values of the
energy-momentum tensor of quantum fields) and quantum gravity (the
full hierarchy of correlation functions retaining complete quantum
coherence [154, 155
]).
The renormalization of the operator is carried out exactly as in the previous case, now
in the perturbed metric
. Note that the stochastic
source
is not dynamical; it is independent of
since it describes the fluctuations of the stress
tensor on the semiclassical background
.
An important property of the Einstein-Langevin
equation is that it is gauge invariant under the change of by
, where
is a stochastic vector field
on the background manifold
. Note that a tensor such as
transforms as
to linear order in the perturbations, where
is the Lie derivative with respect to
. Now, let us write the source tensors in
Equations (14
) and (7
) to the left-hand
sides of these equations. If we substitute
by
in this new version of Equation (14
), we get the same
expression, with
instead of
, plus the Lie
derivative of the combination of tensors which appear on the
left-hand side of the new Equation (7
). This last
combination vanishes when Equation (7
) is satisfied, i.e.,
when the background metric
is a solution of
semiclassical gravity.
A solution of Equation (14) can be formally
written as
. This solution is characterized by the
whole family of its correlation functions. From the statistical
average of this equation we have that
must be a solution of the semiclassical Einstein equation
linearized around the background
; this solution has
been proposed as a test for the validity of the semiclassical
approximation [9
, 10
]. The fluctuations
of the metric around this average are described by the moments of
the stochastic field
. Thus the solutions of the Einstein-Langevin
equation will provide the two-point metric correlation functions
.
We see that whereas the semiclassical theory
depends on the expectation value of the point-defined value of the
stress-energy operator, the stochastic theory carries information
also on the two point correlation of the stress-energy operator. We
should also emphasize that, even if the metric fluctuations appears
classical and stochastic, their origin is quantum not only because
they are induced by the fluctuations of quantum matter, but also
because they are the suitably coarse-grained variables left over
from the quantum gravity fluctuations after some mechanism for
decoherence and classicalization of the metric field [106, 126, 83, 120, 122, 293]. One may, in fact,
derive the stochastic semiclassical theory from a full quantum
theory. This was done via the world-line influence functional
method for a moving charged particle in an electromagnetic field in
quantum electrodynamics [178]. From another
viewpoint, quite independent of whether a classicalization
mechanism is mandatory or implementable, the Einstein-Langevin
equation proves to be a useful tool to compute the symmetrized two
point correlations of the quantum metric perturbations [255
]. This is
illustrated in the linear toy model discussed in [169
], which has features
of some quantum Brownian models [49
, 47, 48].