3.2 Stochastic gravity
The purpose of stochastic gravity is to extend semiclassical theory to account for these fluctuations in a
self-consistent way. A physical observable that describes these fluctuations to lowest order is the noise
kernel bitensor, which is defined through the two-point correlation of the stress-energy operator as
where the curly brackets mean anticommutator, and where
This bitensor can also be written
, or
as we do in Section 5, to emphasize
that it is a tensor with respect to the first two indices at the point
and a tensor with respect to the last
two indices at the point
, but we shall not follow this notation here. The noise kernel is defined in terms
of the unrenormalized stress-tensor operator
on a given background metric
, thus a
regulator is implicitly assumed on the right-hand side of Equation (12). However, for a linear quantum
field, the above kernel – the expectation function of a bitensor – is free of ultraviolet divergences
because the regularized
differs from the renormalized
by the identity
operator times some tensor counterterms (see Equation (7)) so that in the subtraction (13) the
counterterms cancel. Consequently the ultraviolet behavior of
is the same as
that of
, and
can be replaced by the renormalized operator
in
Equation (12); an alternative proof of this result is given in [304
, 305
]. The noise kernel should
be thought of as a distribution function, the limit of coincidence points has meaning only in
the sense of distributions. The bitensor
, or
for short, is real and
positive semi-definite, as a consequence of
being self-adjoint. A simple proof is given
in [208
].
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 (8) 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:
where
means statistical average. The symmetry and positive semi-definite property of the noise
kernel guarantees that the stochastic field tensor
, or
for short, just introduced is
well-defined. Note that this stochastic tensor captures only partially the quantum nature of
the fluctuations of the stress-energy operator as it assumes that cumulants of higher order are
zero.
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 (14) 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:
; so that, 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 (14) that
and
; an alternative proof based on the point-separation method is given
in [304
, 305
], 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:
This equation is in the form of a (semiclassical) Einstein–Langevin equation; it is a dynamical equation for
the metric perturbation
to linear order. It describes the backreaction of the metric to the quantum
fluctuations of the stress-energy tensor of matter fields, and gives a first-order extension to semiclassical
gravity as described by the semiclassical Einstein equation (8).
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; 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 [61] in the Schwinger–Dyson equation hierarchy) to some degree, but one should not expect the
Langevin form to prevail. In this sense we say 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 [187
, 188
].
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 (15) and (8) to the left-hand sides of these
equations. If we substitute
with
in this new version of Equation (15), 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 (8). This last combination vanishes when
Equation (8) is satisfied, i.e., when the background metric
is a solution of semiclassical
gravity.
From the statistical average of Equation (15) 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 [10
, 11
] a point that will be further discussed in
Section 3.3.
The stochastic equation (15) predicts that the gravitational field has stochastic fluctuations over the
background
. This equation is linear in
, thus its solutions can be written as follows,
where
is the solution of the homogeneous equation containing information on the initial
conditions and
is the retarded propagator of Equation (15) with vanishing initial
conditions. Form this we obtain the two-point correlation functions for the metric perturbations:
There are two different contributions to the two-point correlations, which we have distinguished in the
second equality. The first one is connected to the fluctuations of the initial state of the metric perturbations
and we will refer to them as intrinsic fluctuations. The second contribution is proportional
to the noise kernel and is thus connected with the fluctuations of the quantum fields; we will
refer to them as induced fluctuations. To find these two-point stochastic correlation functions
one needs to know the noise kernel
. Explicit expressions of this kernel in terms
of the two-point Wightman functions is given in [258
], expressions based on point-splitting
methods have also been given in [304
, 315
]. Note that the noise kernel should be thought of
as a distribution function, the limit of coincidence points has meaning only in the sense of
distributions.
The two-point stochastic correlation functions for the metric perturbations of Equation (17) satisfy a
very important property. In fact, it can be shown that they correspond exactly to the symmetrized
two-point correlation functions for the quantum metric perturbations in the large
expansion, i.e., the
quantum theory describing the interaction of the gravitational field with
arbitrary free fields and
expanded in powers of
. To leading order for the graviton propagator one finds that
where
is the quantum operator corresponding to the metric perturbations and the statistical
average in Equation (17) for the homogeneous solutions is now taken with respect to the Wigner
distribution that describes the initial quantum state of the metric perturbations. The Lorentz gauge
condition
, as well as an initial condition to completely fix the gauge of the
initial state, should be implicitly understood. Moreover, since there are now
scalar fields, the stochastic
source has been rescaled so that the two-point correlation defined by Equation (14) should
be
times the noise kernel of a single field. This result was implicitly obtained in the
Minkowski background in [259
] where the two-point correlation in the stochastic context was
computed for the linearized metric perturbations. This stochastic correlation exactly agrees with the
symmetrized part of the graviton propagator computed by Tomboulis [348
] in the quantum
context of gravity interacting with
Fermion fields, where the graviton propagator is of
order
. This result can be extended to an arbitrary background in the context of the
large
expansion; a sketch of the proof with explicit details in the Minkowski background
can be found in [203
]. This connection between the stochastic correlations and the quantum
correlations was noted and studied in detail in the context of simpler open quantum systems [66
].
Stochastic gravity goes beyond semiclassical gravity in the following sense. The semiclassical
theory, which is based on the expectation value of the stress-energy tensor, carries information on
the field two-point correlations only, since
is quadratic in the field operator
. The
stochastic theory, on the other hand, is based on the noise kernel 12, which is quartic in the field
operator. However, it does not carry information on the graviton-graviton interaction, which in the
context of the large
expansion gives diagrams of order
. This will be illustrated in
Section 3.3.1. Furthermore, the retarded propagator also gives information on the commutator
so that combining the commutator with the anticommutator, the quantum two-point correlation functions
are determined. Moreover, assuming a Gaussian initial state with vanishing expectation value for the metric
perturbations, any
-point quantum correlation function is determined by the two-point quantum
correlations and thus by the stochastic approach. Consequently, one may regard the Einstein–Langevin
equation as a useful intermediary tool to compute the correlation functions for the quantum metric
perturbations.
We should, however, also emphasize that Langevin-like equations are obtained to describe the
quantum to classical transition in open quantum systems, when quantum decoherence takes
place by coarse-graining of the environment as well as by suitable coarse-graining of the system
variables [101, 127, 144, 146, 150, 374
]. In those cases the stochastic correlation functions describe actual
classical correlations of the system variables. Examples can be found in the case of a moving charged
particle in an electromagnetic field in quantum electrodynamics [219] and in several quantum Brownian
models [64, 65, 66
].