For a free quantum field this theory is robust in the sense that it is selfconsistent and fairly well understood. As long as the gravitational field is assumed to be described by a classical metric, the above semiclassical Einstein equations seems to be the only plausible dynamical equation for this metric: The metric couples to matter fields via the stressenergy tensor, and for a given quantum state the only physically observable cnumber stressenergy tensor that one can construct is the above renormalized expectation value. However, lacking a full quantum gravity theory, the scope and limits of the theory are not so well understood. It is assumed that the semiclassical theory should break down at Planck scales, which is when simple order of magnitude estimates suggest that the quantum effects of gravity should not be ignored, because the energy of a quantum fluctuation in a Planck size region, as determined by the Heisenberg uncertainty principle, is comparable to the gravitational energy of that fluctuation.
The theory is expected to break down when the fluctuations of the stressenergy operator are large [
92]
. A criterion based on the ratio of the fluctuations to the mean was proposed by Kuo and Ford [
194]
(see also work via zetafunction methods [
241,
69]
). This proposal was questioned by Phillips and Hu [
163,
242,
243]
because it does not contain a scale at which the theory is probed or how accurately the theory can be resolved. They suggested the use of a smearing scale or pointseparation distance for integrating over the bitensor quantities, equivalent to a stipulation of the resolution level of measurements; see also the response by Ford [
93,
95]
. A different criterion is recently suggested by Anderson et al. [
10,
9]
based on linear response theory. A partial summary of this issue can be found in our Erice Lectures [
168]
.
3.2 Stochastic gravity
The purpose of stochastic gravity is to extend the semiclassical theory to account for these fluctuations in a selfconsistent 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 stressenergy operator as
$$\begin{array}{c}{N}_{abcd}[g;x,y)=\frac{1}{2}\langle \left\{{\hat{t}}_{ab}\right[g;x),{\hat{t}}_{cd}[g;y\left)\right\}\rangle ,\end{array}$$ 
(11)

where the curly brackets mean anticommutator, and where
$$\begin{array}{c}{\hat{t}}_{ab}[g;x)\equiv {\hat{T}}_{ab}[g;x)\langle {\hat{T}}_{ab}[g;x)\rangle .\end{array}$$ 
(12)

This bitensor can also be written as
${N}_{ab,{c}^{\prime}{d}^{\prime}}[g;x,y)$
, or
${N}_{ab,{c}^{\prime}{d}^{\prime}}(x,y)$
as we do in Section 5 , to emphasize that it is a tensor with respect to the first two indices at the point
$x$
and a tensor with respect to the last two indices at the point
$y$
, but we shall not follow this notation here.
The noise kernel is defined in terms of the unrenormalized stresstensor operator
${\hat{T}}_{ab}[g;x)$
on a given background metric
${g}_{ab}$
, thus a regulator is implicitly assumed on the righthand side of Equation ( 11 ). However, for a linear quantum field the above kernel – the expectation function of a bitensor – is free of ultraviolet divergences because the regularized
${T}_{ab}[g;x)$
differs from the renormalized
${T}_{ab}^{R}[g;x)$
by the identity operator times some tensor counterterms (see Equation ( 6 )), so that in the subtraction ( 12 ) the counterterms cancel. Consequently the ultraviolet behavior of
$\langle {\hat{T}}_{ab}\left(x\right){\hat{T}}_{cd}\left(y\right)\rangle $
is the same as that of
$\langle {\hat{T}}_{ab}\left(x\right)\rangle \langle {\hat{T}}_{cd}\left(y\right)\rangle $
, and
${\hat{T}}_{ab}$
can be replaced by the renormalized operator
${\hat{T}}_{ab}^{R}$
in Equation ( 11 ); an alternative proof of this result is given in [
243,
245]
. 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
${N}_{abcd}[g;x,y)$
, or
${N}_{abcd}(x,y)$
for short, is real and positive semidefinite, as a consequence of
${\hat{T}}_{ab}^{R}$
being selfadjoint. A simple proof is given in [
169]
.
Once the fluctuations of the stressenergy 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
${g}_{ab}$
is a solution of the semiclassical Einstein Equations ( 7 ), and we will write the new metric for the extended theory as
${g}_{ab}+{h}_{ab}$
, where we will assume that
${h}_{ab}$
is a perturbation to the background solution. The renormalized stressenergy operator and the state of the quantum field may now be denoted by
${\hat{T}}_{ab}^{R}[g+h]$
and
$\left\psi \right[g+h]\rangle $
, respectively, and
$\langle {\hat{T}}_{ab}^{R}[g+h]\rangle $
will be the corresponding expectation value.
Let us now introduce a Gaussian stochastic tensor field
${\xi}_{ab}[g;x)$
defined by the following correlators:
$$\begin{array}{c}\langle {\xi}_{ab}[g;x){\rangle}_{s}=0,\langle {\xi}_{ab}[g;x){\xi}_{cd}[g;y){\rangle}_{s}={N}_{abcd}[g;x,y),\end{array}$$ 
(13)

where
$\langle ...{\rangle}_{s}$
means statistical average. The symmetry and positive semidefinite property of the noise kernel guarantees that the stochastic field tensor
${\xi}_{ab}[g,x)$
, or
${\xi}_{ab}\left(x\right)$
for short, just introduced is well defined. Note that this stochastic tensor captures only partially the quantum nature of the fluctuations of the stressenergy operator since 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,
${\nabla}^{a}{\xi}_{ab}[g;x)=0$
. In fact, as a consequence of the conservation of
${\hat{T}}_{ab}^{R}\left[g\right]$
one can see that
${\nabla}_{x}^{a}{N}_{abcd}(x,y)=0$
. Taking the divergence in Equation ( 13 ) one can then show that
$\langle {\nabla}^{a}{\xi}_{ab}{\rangle}_{s}=0$
and
$\langle {\nabla}_{x}^{a}{\xi}_{ab}\left(x\right){\xi}_{cd}\left(y\right){\rangle}_{s}=0$
, so that
${\nabla}^{a}{\xi}_{ab}$
is deterministic and represents with certainty the zero vector field in
$\mathcal{\mathcal{M}}$
.
For a conformal field, i.e., a field whose classical action is conformally invariant,
${\xi}_{ab}$
is traceless:
${g}^{ab}{\xi}_{ab}[g;x)=0$
; 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
${g}^{ab}{\hat{T}}_{ab}^{R}\left[g\right]$
is, in this case, a local cnumber functional of
${g}_{ab}$
times the identity operator, we have that
${g}^{ab}\left(x\right){N}_{abcd}[g;x,y)=0$
.
It then follows from Equation (
13 ) that
$\langle {g}^{ab}{\xi}_{ab}{\rangle}_{s}=0$
and
$\langle {g}^{ab}\left(x\right){\xi}_{ab}\left(x\right){\xi}_{cd}\left(y\right){\rangle}_{s}=0$
; an alternative proof based on the pointseparation method is given in [
243,
245]
(see also Section 5 ). All these properties make it quite natural to incorporate into the Einstein equations the stressenergy fluctuations by using the stochastic tensor
${\xi}_{ab}[g;x)$
as the source of the metric perturbations. Thus we will write the following equation:
$$\begin{array}{c}{G}_{ab}[g+h]+\Lambda ({g}_{ab}+{h}_{ab})2(\alpha {A}_{ab}+\beta {B}_{ab})[g+h]=8\pi G\left(\langle {\hat{T}}_{ab}^{R}[g+h]\rangle +{\xi}_{ab}\left[g\right]\right).\end{array}$$ 
(14)

This equation is in the form of a (semiclassical) Einstein–Langevin equation; it is a dynamical equation for the metric perturbation
${h}_{ab}$
to linear order. It describes the backreaction of the metric to the quantum fluctuations of the stressenergy tensor of matter fields, and gives a first order extension to semiclassical gravity as described by the semiclassical Einstein equation ( 7 ).
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 [
47]
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 energymomentum tensor of quantum fields) and quantum gravity (the full hierarchy of correlation functions retaining complete quantum coherence [
154,
155]
).
The renormalization of the operator
${\hat{T}}_{ab}[g+h]$
is carried out exactly as in the previous case, now in the perturbed metric
${g}_{ab}+{h}_{ab}$
. Note that the stochastic source
${\xi}_{ab}[g;x)$
is not dynamical; it is independent of
${h}_{ab}$
since it describes the fluctuations of the stress tensor on the semiclassical background
${g}_{ab}$
.
An important property of the Einstein–Langevin equation is that it is gauge invariant under the change of
${h}_{ab}$
by
${h}_{ab}^{\prime}={h}_{ab}+{\nabla}_{a}{\zeta}_{b}+{\nabla}_{b}{\zeta}_{a}$
, where
${\zeta}^{a}$
is a stochastic vector field on the background manifold
$\mathcal{\mathcal{M}}$
. Note that a tensor such as
${R}_{ab}[g+h]$
transforms as
${R}_{ab}[g+{h}^{\prime}]={R}_{ab}[g+h]+{\mathcal{\mathcal{L}}}_{\zeta}{R}_{ab}\left[g\right]$
to linear order in the perturbations, where
${\mathcal{\mathcal{L}}}_{\zeta}$
is the Lie derivative with respect to
${\zeta}^{a}$
. Now, let us write the source tensors in Equations ( 14 ) and ( 7 ) to the lefthand sides of these equations. If we substitute
$h$
by
${h}^{\prime}$
in this new version of Equation ( 14 ), we get the same expression, with
$h$
instead of
${h}^{\prime}$
, plus the Lie derivative of the combination of tensors which appear on the lefthand side of the new Equation ( 7 ). This last combination vanishes when Equation ( 7 ) is satisfied, i.e., when the background metric
${g}_{ab}$
is a solution of semiclassical gravity.
A solution of Equation (
14 ) can be formally written as
${h}_{ab}\left[\xi \right]$
. This solution is characterized by the whole family of its correlation functions. From the statistical average of this equation we have that
${g}_{ab}+\langle {h}_{ab}{\rangle}_{s}$
must be a solution of the semiclassical Einstein equation linearized around the background
${g}_{ab}$
; this solution has been proposed as a test for the validity of the semiclassical approximation [
10,
9]
. The fluctuations of the metric around this average are described by the moments of the stochastic field
${h}_{ab}^{s}\left[\xi \right]={h}_{ab}\left[\xi \right]\langle {h}_{ab}{\rangle}_{s}$
. Thus the solutions of the Einstein–Langevin equation will provide the twopoint metric correlation functions
$\langle {h}_{ab}^{s}\left(x\right){h}_{cd}^{s}\left(y\right){\rangle}_{s}$
.
We see that whereas the semiclassical theory depends on the expectation value of the pointdefined value of the stressenergy operator, the stochastic theory carries information also on the two point correlation of the stressenergy 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 coarsegrained 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 worldline 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 [
51,
49,
50]
.
4 The Einstein–Langevin Equation: Functional Approach
The Einstein–Langevin equation ( 14 ) may also be derived by a method based on functional techniques [
208]
. Here we will summarize these techniques starting with semiclassical gravity.
In semiclassical gravity functional methods were used to study the backreaction of quantum fields in cosmological models [
123,
90,
129]
. The primary advantage of the effective action approach is, in addition to the wellknown fact that it is easy to introduce perturbation schemes like loop expansion and nPI formalisms, that it yields a fully selfconsistent solution. For a general discussion in the semiclassical context of these two approaches, equation of motion versus effective action, see, e.g., the work of Hu and Parker (1978) versus Hartle and Hu (1979) in [
203,
115,
158,
159,
124,
3,
4]
. See also comments in Sec. 5.6 of [
169]
on the black hole backreaction problem comparing the approach by York et al. [
297,
298,
299]
versus that of Sinha, Raval, and Hu [
264]
.
The well known inout effective action method treated in textbooks, however, led to equations of motion which were not real because they were tailored to compute transition elements of quantum operators rather than expectation values. The correct technique to use for the backreaction problem is the Schwinger–Keldysh closedtimepath (CTP) or `inin' effective action [
257,
11,
184,
66,
272,
42,
70]
.
These techniques were adapted to the gravitational context [
76,
181,
40,
182,
236,
57]
and applied to different problems in cosmology. One could deduce the semiclassical Einstein equation from the CTP effective action for the gravitational field (at tree level) with quantum matter fields.
Furthermore, in this case the CTP functional formalism turns out to be related [
272,
44,
58,
201,
112,
54,
55,
216,
196,
208,
206]
to the influence functional formalism of Feynman and Vernon [
89]
, since the full quantum system may be understood as consisting of a distinguished subsystem (the “system” of interest) interacting with the remaining degrees of freedom (the environment). Integrating out the environment variables in a CTP path integral yields the influence functional, from which one can define an effective action for the dynamics of the system [
44,
167,
156,
112]
. This approach to semiclassical gravity is motivated by the observation [
148]
that in some open quantum systems classicalization and decoherence [
303,
304,
305,
306,
180,
33,
279,
307,
109]
on the system may be brought about by interaction with an environment, the environment being in this case the matter fields and some “highmomentum” gravitational modes [
188,
119,
228,
149,
36,
37,
160,
293]
. Unfortunately, since the form of a complete quantum theory of gravity interacting with matter is unknown, we do not know what these “highmomentum” gravitational modes are. Such a fundamental quantum theory might not even be a field theory, in which case the metric and scalar fields would not be fundamental objects [
154]
. Thus, in this case, we cannot attempt to evaluate the influence action of Feynman and Vernon starting from the fundamental quantum theory and performing the path integrations in the environment variables.
Instead, we introduce the influence action for an effective quantum field theory of gravity and matter [
78,
77,
79,
80,
263,
237,
238]
, in which such “highmomentum” gravitational modes are assumed to have already been “integrated out.”
4.1 Influence action for semiclassical gravity
Let us formulate semiclassical gravity in this functional framework. Adopting the usual procedure of effective field theories [
289,
290,
78,
77,
79,
80,
48]
, one has to take the effective action for the metric and the scalar field of the most general local form compatible with general covariance:
$S[g,\phi ]\equiv {S}_{g}\left[g\right]+{S}_{m}[g,\phi ]+...$
, where
${S}_{g}\left[g\right]$
and
${S}_{m}[g,\phi ]$
are given by Equations ( 8 ) and ( 1 ), respectively, and the dots stand for terms of order higher than two in the curvature and in the number of derivatives of the scalar field. Here, we shall neglect the higher order terms as well as selfinteraction terms for the scalar field. The second order terms are necessary to renormalize oneloop ultraviolet divergences of the scalar field stressenergy tensor, as we have already seen. Since
$\mathcal{\mathcal{M}}$
is a globally hyperbolic manifold, we can foliate it by a family of
$t=const.$
Cauchy hypersurfaces
${\Sigma}_{t}$
, and we will indicate the initial and final times by
${t}_{i}$
and
${t}_{f}$
, respectively.
The
influence functional corresponding to the action ( 1 ) describing a scalar field in a spacetime (coupled to a metric field) may be introduced as a functional of two copies of the metric, denoted by
${g}_{ab}^{+}$
and
${g}_{ab}^{}$
, which coincide at some final time
$t={t}_{f}$
. Let us assume that, in the quantum effective theory, the state of the full system (the scalar and the metric fields) in the Schrödinger picture at the initial time
$t={t}_{i}$
can be described by a density operator which can be written as the tensor product of two operators on the Hilbert spaces of the metric and of the scalar field. Let
${\rho}_{i}\left({t}_{i}\right)\equiv {\rho}_{i}\left[{\phi}_{+}\left({t}_{i}\right),{\phi}_{}\left({t}_{i}\right)\right]$
be the matrix element of the density operator
${\hat{\rho}}^{S}\left({t}_{i}\right)$
describing the initial state of the scalar field. The Feynman–Vernon influence functional is defined as the following path integral over the two copies of the scalar field:
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{IF}\left[{g}^{\pm}\right]\equiv \int \mathcal{D}{\phi}_{+}\mathcal{D}{\phi}_{}{\rho}_{i}\left({t}_{i}\right)\delta \left[{\phi}_{+}\left({t}_{f}\right){\phi}_{}\left({t}_{f}\right)\right]{e}^{i\left({S}_{m}[{g}^{+},{\phi}_{+}]{S}_{m}[{g}^{},{\phi}_{}]\right)}.\end{array}$$ 
(15)

Alternatively, the above double path integral can be rewritten as a CTP integral, namely, as a single path integral in a complex time contour with two different time branches, one going forward in time from
${t}_{i}$
to
${t}_{f}$
, and the other going backward in time from
${t}_{f}$
to
${t}_{i}$
(in practice one usually takes
${t}_{i}\to \infty $
). From this influence functional, the influence action
${S}_{IF}[{g}^{+},{g}^{}]$
, or
${S}_{IF}\left[{g}^{\pm}\right]$
for short, defined by
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{IF}\left[{g}^{\pm}\right]\equiv {e}^{i{S}_{IF}\left[{g}^{\pm}\right]},\end{array}$$ 
(16)

carries all the information about the environment (the matter fields) relevant to the system (the gravitational field). Then we can define the CTP effective action for the gravitational field,
${S}_{eff}\left[{g}^{\pm}\right]$
, as
$$\begin{array}{c}{S}_{eff}\left[{g}^{\pm}\right]\equiv {S}_{g}\left[{g}^{+}\right]{S}_{g}\left[{g}^{}\right]+{S}_{IF}\left[{g}^{\pm}\right].\end{array}$$ 
(17)

This is the effective action for the classical gravitational field in the CTP formalism. However, since the gravitational field is treated only at the tree level, this is also the effective classical action from which the classical equations of motion can be derived.
Expression (
15 ) contains ultraviolet divergences and must be regularized. We shall assume that dimensional regularization can be applied, that is, it makes sense to dimensionally continue all the quantities that appear in Equation ( 15 ). For this we need to work with the
$n$
dimensional actions corresponding to
${S}_{m}$
in Equation ( 15 ) and
${S}_{g}$
in Equation ( 8 ). For example, the parameters
$G$
,
$\Lambda $
,
$\alpha $
, and
$\beta $
of Equation ( 8 ) are the bare parameters
${G}_{B}$
,
${\Lambda}_{B}$
,
${\alpha}_{B}$
, and
${\beta}_{B}$
, and in
${S}_{g}\left[g\right]$
, instead of the square of the Weyl tensor in Equation ( 8 ), one must use
$\frac{2}{3}({R}_{abcd}{R}^{abcd}{R}_{ab}{R}^{ab})$
, which by the Gauss–Bonnet theorem leads to the same equations of motion as the action ( 8 ) when
$n=4$
. The form of
${S}_{g}$
in
$n$
dimensions is suggested by the Schwinger–DeWitt analysis of the ultraviolet divergences in the matter stressenergy tensor using dimensional regularization. One can then write the Feynman–Vernon effective action
${S}_{eff}\left[{g}^{\pm}\right]$
in Equation ( 17 ) in a form suitable for dimensional regularization. Since both
${S}_{m}$
and
${S}_{g}$
contain second order derivatives of the metric, one should also add some boundary terms [
284,
167]
. The effect of these terms is to cancel out the boundary terms which appear when taking variations of
${S}_{eff}\left[{g}^{\pm}\right]$
keeping the value of
${g}_{ab}^{+}$
and
${g}_{ab}^{}$
fixed at
${\Sigma}_{{t}_{i}}$
and
${\Sigma}_{{t}_{f}}$
. Alternatively, in order to obtain the equations of motion for the metric in the semiclassical regime, we can work with the action terms without boundary terms and neglect all boundary terms when taking variations with respect to
${g}_{ab}^{\pm}$
. From now on, all the functional derivatives with respect to the metric will be understood in this sense.
The semiclassical Einstein equation (
7 ) can now be derived. Using the definition of the stressenergy tensor
${T}^{ab}\left(x\right)=(2/\sqrt{g})\delta {S}_{m}/\delta {g}_{ab}$
and the definition of the influence functional, Equations ( 15 ) and ( 16 ), we see that
$$\begin{array}{c}\langle {\hat{T}}^{ab}[g;x)\rangle ={\frac{2}{\sqrt{g\left(x\right)}}\frac{\delta {S}_{IF}\left[{g}^{\pm}\right]}{\delta {g}_{ab}^{+}\left(x\right)}}_{{g}^{\pm}=g},\end{array}$$ 
(18)

where the expectation value is taken in the
$n$
dimensional spacetime generalization of the state described by
${\hat{\rho}}^{S}\left({t}_{i}\right)$
. Therefore, differentiating
${S}_{eff}\left[{g}^{\pm}\right]$
in Equation ( 17 ) with respect to
${g}_{ab}^{+}$
, and then setting
${g}_{ab}^{+}={g}_{ab}^{}={g}_{ab}$
, we get the semiclassical Einstein equation in
$n$
dimensions. This equation is then renormalized by absorbing the divergences in the regularized
$\langle {\hat{T}}^{ab}\left[g\right]\rangle $
into the bare parameters. Taking the limit
$n\to 4$
we obtain the physical semiclassical Einstein equation ( 7 ).
4.2 Influence action for stochastic gravity
In the spirit of the previous derivation of the Einstein–Langevin equation, we now seek a dynamical equation for a linear perturbation
${h}_{ab}$
to the semiclassical metric
${g}_{ab}$
, solution of Equation ( 7 ). Strictly speaking, if we use dimensional regularization we must consider the
$n$
dimensional version of that equation. From the results just described, if such an equation were simply a linearized semiclassical Einstein equation, it could be obtained from an expansion of the effective action
${S}_{eff}[g+{h}^{\pm}]$
. In particular, since, from Equation ( 18 ), we have that
$$\begin{array}{c}\langle {\hat{T}}^{ab}[g+h;x)\rangle ={\frac{2}{\sqrt{det(g+h)\left(x\right)}}\frac{\delta {S}_{IF}[g+{h}^{\pm}]}{\delta {h}_{ab}^{+}\left(x\right)}}_{{h}^{\pm}=h},\end{array}$$ 
(19)

the expansion of
$\langle {\hat{T}}^{ab}[g+h]\rangle $
to linear order in
${h}_{ab}$
can be obtained from an expansion of the influence action
${S}_{IF}[g+{h}^{\pm}]$
up to second order in
${h}_{ab}^{\pm}$
.
To perform the expansion of the influence action, we have to compute the first and second order functional derivatives of
${S}_{IF}[g+{h}^{\pm}]$
and then set
${h}_{ab}^{+}={h}_{ab}^{}={h}_{ab}$
. If we do so using the path integral representation ( 15 ), we can interpret these derivatives as expectation values of operators.
The relevant second order derivatives are
$$\begin{array}{c}\begin{array}{ccc}{\frac{4}{\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}}\frac{{\delta}^{2}{S}_{IF}[g+{h}^{\pm}]}{\delta {h}_{ab}^{+}\left(x\right)\delta {h}_{cd}^{+}\left(y\right)}}_{{h}^{\pm}=h}& =& {H}_{S}^{abcd}[g;x,y){K}^{abcd}[g;x,y)+i{N}^{abcd}[g;x,y),\\ {\frac{4}{\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}}\frac{{\delta}^{2}{S}_{IF}\left[{g}^{\pm}\right]}{\delta {h}_{ab}^{+}\left(x\right)\delta {h}_{cd}^{}\left(y\right)}}_{{h}^{\pm}=h}& =& {H}_{A}^{abcd}[g;x,y)i{N}^{abcd}[g;x,y),\end{array}\end{array}$$ 
(20)

where
$$\begin{array}{ccc}{N}^{abcd}[g;x,y)& \equiv & \frac{1}{2}\langle \left\{{\hat{t}}^{ab}[g;x),{\hat{t}}^{cd}[g;y)\right\}\rangle ,\end{array}$$  
$$\begin{array}{ccc}{H}_{S}^{abcd}[g;x,y)& \equiv & Im\langle {T}^{*}\left({\hat{T}}^{ab}[g;x){\hat{T}}^{cd}[g;y)\right)\rangle ,\end{array}$$  
$$\begin{array}{ccc}{H}_{A}^{abcd}[g;x,y)& \equiv & \frac{i}{2}\langle \left[{\hat{T}}^{ab}[g;x),{\hat{T}}^{cd}[g;y)\right]\rangle ,\end{array}$$  
$$\begin{array}{ccc}{K}^{abcd}[g;x,y)& \equiv & \frac{4}{\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}}\langle {\frac{{\delta}^{2}{S}_{m}[g+h,\phi ]}{\delta {h}_{ab}\left(x\right)\delta {h}_{cd}\left(y\right)}}_{\phi =\hat{\phi}}\rangle ,\end{array}$$  
with
${\hat{t}}^{ab}$
defined in Equation ( 12 );
$[,]$
denotes the commutator and
$\{,\}$
the anticommutator.
Here we use a Weyl ordering prescription for the operators. The symbol
${T}^{*}$
denotes the following ordered operations: First, time order the field operators
$\hat{\phi}$
and then apply the derivative operators which appear in each term of the product
${T}^{ab}\left(x\right){T}^{cd}\left(y\right)$
, where
${T}^{ab}$
is the functional ( 3 ). This
${T}^{*}$
“time ordering” arises because we have path integrals containing products of derivatives of the field, which can be expressed as derivatives of the path integrals which do not contain such derivatives. Notice, from their definitions, that all the kernels which appear in expressions ( 20 ) are real and also
${H}_{A}^{abcd}$
is free of ultraviolet divergences in the limit
$n\to 4$
.
From Equation (
18 ) and ( 20 ), since
${S}_{IF}[g,g]=0$
and
${S}_{IF}[{g}^{},{g}^{+}]={S}_{IF}^{*}[{g}^{+},{g}^{}]$
, we can write the expansion for the influence action
${S}_{IF}[g+{h}^{\pm}]$
around a background metric
${g}_{ab}$
in terms of the previous kernels. Taking into account that these kernels satisfy the symmetry relations
$$\begin{array}{c}{H}_{S}^{abcd}(x,y)={H}_{S}^{cdab}(y,x),{H}_{A}^{abcd}(x,y)={H}_{A}^{cdab}(y,x),{K}^{abcd}(x,y)={K}^{cdab}(y,x),\end{array}$$ 
(21)

and introducing the new kernel
$$\begin{array}{c}{H}^{abcd}(x,y)\equiv {H}_{S}^{abcd}(x,y)+{H}_{A}^{abcd}(x,y),\end{array}$$ 
(22)

the expansion of
${S}_{IF}$
can be finally written as
$$\begin{array}{ccc}{S}_{IF}[g+{h}^{\pm}]& =& \frac{1}{2}\int {d}^{4}x\sqrt{g\left(x\right)}\langle {\hat{T}}^{ab}[g;x)\rangle \left[{h}_{ab}\left(x\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & \frac{1}{8}\int {d}^{4}x{d}^{4}y\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}\left[{h}_{ab}\left(x\right)\right]\left({H}^{abcd}[g;x,y)+{K}^{abcd}[g;x,y)\right)\left\{{h}_{cd}\left(y\right)\right\}\end{array}$$  
$$\begin{array}{ccc}& & +\frac{i}{8}\int {d}^{4}x{d}^{4}y\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}\left[{h}_{ab}\left(x\right)\right]{N}^{abcd}[g;x,y)\left[{h}_{cd}\left(y\right)\right]+\mathcal{O}\left({h}^{3}\right),\end{array}$$ 
(23)

where we have used the notation
$$\begin{array}{c}\left[{h}_{ab}\right]\equiv {h}_{ab}^{+}{h}_{ab}^{},\left\{{h}_{ab}\right\}\equiv {h}_{ab}^{+}+{h}_{ab}^{}.\end{array}$$ 
(24)

From Equations ( 23 ) and ( 19 ) it is clear that the imaginary part of the influence action does not contribute to the perturbed semiclassical Einstein equation (the expectation value of the stressenergy tensor is real), however, as it depends on the noise kernel, it contains information on the fluctuations of the operator
${\hat{T}}^{ab}\left[g\right]$
.
We are now in a position to carry out the derivation of the semiclassical Einstein–Langevin equation. The procedure is well known [
44,
167,
58,
110,
26,
296,
246]
: It consists of deriving a new “stochastic” effective action from the observation that the effect of the imaginary part of the influence action ( 23 ) on the corresponding influence functional is equivalent to the averaged effect of the stochastic source
${\xi}^{ab}$
coupled linearly to the perturbations
${h}_{ab}^{\pm}$
. This observation follows from the identity first invoked by Feynman and Vernon for such purpose:
$$\begin{array}{ccc}exp\left(\frac{1}{8}\int {d}^{4}x{d}^{4}y\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}\left[{h}_{ab}\left(x\right)\right]{N}^{abcd}(x,y)\left[{h}_{cd}\left(y\right)\right]\right)=& & \end{array}$$  
$$\begin{array}{ccc}\int \mathcal{D}\xi \mathcal{P}\left[\xi \right]exp\left(\frac{i}{2}\int {d}^{4}x\sqrt{g\left(x\right)}{\xi}^{ab}\left(x\right)\left[{h}_{ab}\left(x\right)\right]\right),& & \end{array}$$ 
(25)

where
$\mathcal{P}\left[\xi \right]$
is the probability distribution functional of a Gaussian stochastic tensor
${\xi}^{ab}$
characterized by the correlators ( 13 ) with
${N}^{abcd}$
given by Equation ( 11 ), and where the path integration measure is assumed to be a scalar under diffeomorphisms of
$(\mathcal{\mathcal{M}},{g}_{ab})$
. The above identity follows from the identification of the righthand side of Equation ( 25 ) with the characteristic functional for the stochastic field
${\xi}^{ab}$
. The probability distribution functional for
${\xi}^{ab}$
is explicitly given by
$$\begin{array}{c}\mathcal{P}\left[\xi \right]=det(2\pi N{)}^{1/2}exp\left[\frac{1}{2}\int {d}^{4}x{d}^{4}y\sqrt{g\left(x\right)}\sqrt{g\left(y\right)}{\xi}^{ab}\left(x\right){N}_{abcd}^{1}(x,y){\xi}^{cd}\left(y\right)\right].\end{array}$$ 
(26)

We may now introduce the stochastic effective action as
$$\begin{array}{c}{S}_{eff}^{s}[g+{h}^{\pm},\xi ]\equiv {S}_{g}[g+{h}^{+}]{S}_{g}[g+{h}^{}]+{S}_{IF}^{s}[g+{h}^{\pm},\xi ],\end{array}$$ 
(27)

where the “stochastic” influence action is defined as
$$\begin{array}{c}{S}_{IF}^{s}[g+{h}^{\pm},\xi ]\equiv Re{S}_{IF}[g+{h}^{\pm}]+\frac{1}{2}\int {d}^{4}x\sqrt{g\left(x\right)}{\xi}^{ab}\left(x\right)\left[{h}_{ab}\left(x\right)\right]+\mathcal{O}\left({h}^{3}\right).\end{array}$$ 
(28)

Note that, in fact, the influence functional can now be written as a statistical average over
${\xi}^{ab}$
:
$${\mathcal{\mathcal{F}}}_{IF}[g+{h}^{\pm}]={\langle exp\left(i{S}_{IF}^{s}[g+{h}^{\pm},\xi ]\right)\rangle}_{s}.$$
The stochastic equation of motion for
${h}_{ab}$
reads
$$\begin{array}{c}{\frac{\delta {S}_{eff}^{s}[g+{h}^{\pm},\xi ]}{\delta {h}_{ab}^{+}\left(x\right)}}_{{h}^{\pm}=h}=0,\end{array}$$ 
(29)

which is the Einstein–Langevin equation ( 14 ); notice that only the real part of
${S}_{IF}$
contributes to the expectation value ( 19 ). To be precise, we get first the regularized
$n$
dimensional equations with the bare parameters, with the tensor
${A}^{ab}$
replaced by
$\frac{2}{3}{D}^{ab}$
, where
$$\begin{array}{ccc}{D}^{ab}& \equiv & \frac{1}{\sqrt{g}}\frac{\delta}{\delta {g}_{ab}}\int {d}^{n}x\sqrt{g}\left({R}_{cdef}{R}^{cdef}{R}_{cd}{R}^{cd}\right)\end{array}$$  
$$\begin{array}{ccc}& =& \frac{{g}^{ab}}{2}\left({R}_{cdef}{R}^{cdef}{R}_{cd}{R}^{cd}+\square R\right)2{R}^{acde}{R}_{cde}^{b}2{R}^{acbd}{R}_{cd}+4{R}^{ac}{{R}_{c}}^{b}3\square {R}^{ab}+{\nabla}^{a}{\nabla}^{b}R.\end{array}$$  
$$\begin{array}{ccc}& & \end{array}$$ 
(30)

Of course, when
$n=4$
these tensors are related,
${A}^{ab}=\frac{2}{3}{D}^{ab}$
. After that we renormalize and take the limit
$n\to 4$
to obtain the Einstein–Langevin equations in the physical spacetime.
4.3 Explicit form of the Einstein–Langevin equation
We can write the Einstein–Langevin equation in a more explicit form by working out the expansion of
$\langle {\hat{T}}^{ab}[g+h]\rangle $
up to linear order in the perturbation
${h}_{ab}$
. From Equation ( 19 ), we see that this expansion can be easily obtained from Equation ( 23 ). The result is
$$\begin{array}{c}\langle {\hat{T}}_{n}^{ab}[g+h;x)\rangle =\langle {\hat{T}}_{n}^{ab}[g,x)\rangle +\langle {\hat{T}}_{n}^{\left(1\right)ab}[g,h;x)\rangle \frac{1}{2}\int {d}^{n}y\sqrt{g\left(y\right)}{H}_{n}^{abcd}[g;x,y){h}_{cd}\left(y\right)+\mathcal{O}\left({h}^{2}\right).\end{array}$$ 
(31)

Here we use a subscript
$n$
on a given tensor to indicate that we are explicitly working in
$n$
dimensions, as we use dimensional regularization, and we also use the superindex
${}^{\left(1\right)}$
to generally indicate that the tensor is the first order correction, linear in
${h}_{ab}$
, in a perturbative expansion around the background
${g}_{ab}$
.
Using the Klein–Gordon equation (
2 ), and expressions ( 3 ) for the stressenergy tensor and the corresponding operator, we can write
$$\begin{array}{c}{\hat{T}}_{n}^{\left(1\right)ab}[g,h]=\left(\frac{1}{2}{g}^{ab}{h}_{cd}{\delta}_{c}^{a}{h}_{d}^{b}{\delta}_{c}^{b}{h}_{d}^{a}\right){\hat{T}}_{n}^{cd}\left[g\right]+{\mathcal{\mathcal{F}}}^{ab}[g,h]{\hat{\phi}}_{n}^{2}\left[g\right],\end{array}$$ 
(32)

where
${\mathcal{\mathcal{F}}}^{ab}[g;h]$
is the differential operator
$$\begin{array}{ccc}{\mathcal{\mathcal{F}}}^{ab}& \equiv & \left(\xi \frac{1}{4}\right)\left({h}^{ab}\frac{1}{2}{g}^{ab}{h}_{c}^{c}\right)\square \end{array}$$  
$$\begin{array}{ccc}& & +\frac{\xi}{2}[{\nabla}^{c}{\nabla}^{a}{h}_{c}^{b}+{\nabla}^{c}{\nabla}^{b}{h}_{c}^{a}\square {h}^{ab}{\nabla}^{a}{\nabla}^{b}{h}_{c}^{c}{g}^{ab}{\nabla}^{c}{\nabla}^{d}{h}_{cd}+{g}^{ab}\square {h}_{c}^{c}\end{array}$$  
$$\begin{array}{ccc}& & +\left({\nabla}^{a}{h}_{c}^{b}+{\nabla}^{b}{h}_{c}^{a}{\nabla}_{c}{h}^{ab}2{g}^{ab}{\nabla}^{d}{h}_{cd}+{g}^{ab}{\nabla}_{c}{h}_{d}^{d}\right){\nabla}^{c}{g}^{ab}{h}_{cd}{\nabla}^{c}{\nabla}^{d}].\end{array}$$ 
(33)

It is understood that indices are raised with the background inverse metric
${g}^{ab}$
, and that all the covariant derivatives are associated to the metric
${g}_{ab}$
.
Substituting Equation (
31 ) into the
$n$
dimensional version of the Einstein–Langevin Equation ( 14 ), taking into account that
${g}_{ab}$
satisfies the semiclassical Einstein equation ( 7 ), and substituting expression ( 32 ), we can write the Einstein–Langevin equation in dimensional regularization as
$$\begin{array}{ccc}& & \frac{1}{8\pi {G}_{B}}\left[{G}^{\left(1\right)ab}\frac{1}{2}{g}^{ab}{G}^{cd}{h}_{cd}+{G}^{ac}{h}_{c}^{b}+{G}^{bc}{h}_{c}^{a}+{\Lambda}_{B}\left({h}^{ab}\frac{1}{2}{g}^{ab}{h}_{c}^{c}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & \frac{4{\alpha}_{B}}{3}\left({D}^{\left(1\right)ab}\frac{1}{2}{g}^{ab}{D}^{cd}{h}_{cd}+{D}^{ac}{h}_{c}^{b}+{D}^{bc}{h}_{c}^{a}\right)\end{array}$$  
$$\begin{array}{ccc}& & 2{\beta}_{B}\left({B}^{\left(1\right)ab}\frac{1}{2}{g}^{ab}{B}^{cd}{h}_{cd}+{B}^{ac}{h}_{c}^{b}+{B}^{bc}{h}_{c}^{a}\right)\end{array}$$  
$$\begin{array}{ccc}& & {\mu}^{(n4)}{\mathcal{\mathcal{F}}}_{x}^{ab}\langle {\hat{\phi}}_{n}^{2}[g;x)\rangle +\frac{1}{2}\int {d}^{n}y\sqrt{g\left(y\right)}{\mu}^{(n4)}{H}_{n}^{abcd}[g;x,y){h}_{cd}\left(y\right)\end{array}$$  
$$\begin{array}{ccc}& =& {\mu}^{(n4)}{\xi}_{n}^{ab},\end{array}$$ 
(34)

where the tensors
${G}^{ab}$
,
${D}^{ab}$
, and
${B}^{ab}$
are computed from the semiclassical metric
${g}_{ab}$
, and where we have omitted the functional dependence on
${g}_{ab}$
and
${h}_{ab}$
in
${G}^{\left(1\right)ab}$
,
${D}^{\left(1\right)ab}$
,
${B}^{\left(1\right)ab}$
, and
${\mathcal{\mathcal{F}}}^{ab}$
to simplify the notation. The parameter
$\mu $
is a mass scale which relates the dimensions of the physical field
$\phi $
with the dimensions of the corresponding field in
$n$
dimensions,
${\phi}_{n}={\mu}^{(n4)/2}\phi $
. Notice that, in Equation ( 34 ), all the ultraviolet divergences in the limit
$n\to 4$
, which must be removed by renormalization of the coupling constants, are in
$\langle {\hat{\phi}}_{n}^{2}\left(x\right)\rangle $
and the symmetric part
${H}_{{S}_{n}}^{abcd}(x,y)$
of the kernel
${H}_{n}^{abcd}(x,y)$
, whereas the kernels
${N}_{n}^{abcd}(x,y)$
and
${H}_{{A}_{n}}^{abcd}(x,y)$
are free of ultraviolet divergences. If we introduce the bitensor
${F}_{n}^{abcd}[g;x,y)$
defined by
$$\begin{array}{c}{F}_{n}^{abcd}[g;x,y)\equiv \langle {\hat{t}}_{n}^{ab}[g;x){\hat{t}}_{n}^{\rho \sigma}[g;y)\rangle ,\end{array}$$ 
(35)

where
${\hat{t}}^{ab}$
is defined by Equation ( 12 ), then the kernels
$N$
and
${H}_{A}$
can be written as
$$\begin{array}{c}{N}_{n}^{abcd}[g;x,y)=Re{F}_{n}^{abcd}[g;x,y),{H}_{{A}_{n}}^{abcd}[g;x,y)=Im{F}_{n}^{abcd}[g;x,y),\end{array}$$ 
(36)

where we have used that
$$2\langle {\hat{t}}^{ab}\left(x\right){\hat{t}}^{cd}\left(y\right)\rangle =\langle \left\{{\hat{t}}^{ab}\right(x),{\hat{t}}^{cd}(y\left)\right\}\rangle +\langle \left[{\hat{t}}^{ab}\right(x),{\hat{t}}^{cd}(y\left)\right]\rangle ,$$
and the fact that the first term on the righthand side of this identity is real, whereas the second one is pure imaginary. Once we perform the renormalization procedure in Equation ( 34 ), setting
$n=4$
will yield the physical Einstein–Langevin equation. Due to the presence of the kernel
${H}_{n}^{abcd}(x,y)$
, this equation will be usually nonlocal in the metric perturbation. In Section 6 we will carry out an explicit evaluation of the physical Einstein–Langevin equation which will illustrate the procedure.
4.3.1 The kernels for the vacuum state
When the expectation values in the Einstein–Langevin equation are taken in a vacuum state
$0\rangle $
, such as, for instance, an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the Wightman and Feynman functions, defined as
$$\begin{array}{c}{G}_{n}^{+}[g;x,y)\equiv \langle 0\left{\hat{\phi}}_{n}\right[g;x\left){\hat{\phi}}_{n}\right[g;y\left)\right0\rangle ,i{G}_{{F}_{n}}[g;x,y)\equiv \langle 0T\left({\hat{\phi}}_{n}[g;x){\hat{\phi}}_{n}[g;y)\right)0\rangle .\end{array}$$ 
(37)

These expressions for the kernels in the Einstein–Langevin equation will be very useful for explicit calculations. To simplify the notation, we omit the functional dependence on the semiclassical metric
${g}_{ab}$
, which will be understood in all the expressions below.
From Equations (
36 ), we see that the kernels
${N}_{n}^{abcd}(x,y)$
and
${H}_{{A}_{n}}^{abcd}(x,y)$
are the real and imaginary parts, respectively, of the bitensor
${F}_{n}^{abcd}(x,y)$
. From the expression ( 4 ) we see that the stressenergy operator
${\hat{T}}_{n}^{ab}$
can be written as a sum of terms of the form
$\left\{{\mathcal{A}}_{x}{\hat{\phi}}_{n}\left(x\right),{\mathcal{\mathcal{B}}}_{x}{\hat{\phi}}_{n}\left(x\right)\right\}$
, where
${\mathcal{A}}_{x}$
and
${\mathcal{\mathcal{B}}}_{x}$
are some differential operators. It then follows that we can express the bitensor
${F}_{n}^{abcd}(x,y)$
in terms of the Wightman function as
$$\begin{array}{ccc}{F}_{n}^{abcd}(x,y)& =& {\nabla}_{x}^{a}{\nabla}_{y}^{c}{G}_{n}^{+}(x,y){\nabla}_{x}^{b}{\nabla}_{y}^{d}{G}_{n}^{+}(x,y)+{\nabla}_{x}^{a}{\nabla}_{y}^{d}{G}_{n}^{+}(x,y){\nabla}_{x}^{b}{\nabla}_{y}^{c}{G}_{n}^{+}(x,y)\end{array}$$  
$$\begin{array}{ccc}& & +2{\mathcal{D}}_{x}^{ab}\left({\nabla}_{y}^{c}{G}_{n}^{+}(x,y){\nabla}_{y}^{d}{G}_{n}^{+}(x,y)\right)+2{\mathcal{D}}_{y}^{cd}\left({\nabla}_{x}^{a}{G}_{n}^{+}(x,y){\nabla}_{x}^{b}{G}_{n}^{+}(x,y)\right)\end{array}$$  
$$\begin{array}{ccc}& & +2{\mathcal{D}}_{x}^{ab}{\mathcal{D}}_{y}^{cd}\left({G}_{n}^{+2}(x,y)\right),\end{array}$$ 
(38)

where
${\mathcal{D}}_{x}^{ab}$
is the differential operator ( 5 ). From this expression and the relations ( 36 ), we get expressions for the kernels
${N}_{n}$
and
${H}_{{A}_{n}}$
in terms of the Wightman function
${G}_{n}^{+}(x,y)$
.
Similarly the kernel
${H}_{{S}_{n}}^{abcd}(x,y)$
can be written in terms of the Feynman function as
$$\begin{array}{ccc}{H}_{{S}_{n}}^{abcd}(x,y)& =& Im[{\nabla}_{x}^{a}{\nabla}_{y}^{c}{G}_{{F}_{n}}(x,y){\nabla}_{x}^{b}{\nabla}_{y}^{d}{G}_{{F}_{n}}(x,y)+{\nabla}_{x}^{a}{\nabla}_{y}^{d}{G}_{{F}_{n}}(x,y){\nabla}_{x}^{b}{\nabla}_{y}^{c}{G}_{{F}_{n}}(x,y)\end{array}$$  
$$\begin{array}{ccc}& & {g}^{ab}\left(x\right){\nabla}_{x}^{e}{\nabla}_{y}^{c}{G}_{{F}_{n}}(x,y){\nabla}_{e}^{x}{\nabla}_{y}^{d}{G}_{{F}_{n}}(x,y)\end{array}$$  
$$\begin{array}{ccc}& & {g}^{cd}\left(y\right){\nabla}_{x}^{a}{\nabla}_{y}^{e}{G}_{{F}_{n}}(x,y){\nabla}_{x}^{b}{\nabla}_{e}^{y}{G}_{{F}_{n}}(x,y)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{2}{g}^{ab}\left(x\right){g}^{cd}\left(y\right){\nabla}_{x}^{e}{\nabla}_{y}^{f}{G}_{{F}_{n}}(x,y){\nabla}_{e}^{x}{\nabla}_{f}^{y}{G}_{{F}_{n}}(x,y)\end{array}$$  
$$\begin{array}{ccc}& & +{\mathcal{K}}_{x}^{ab}\left(2{\nabla}_{y}^{c}{G}_{{F}_{n}}(x,y){\nabla}_{y}^{d}{G}_{{F}_{n}}(x,y){g}^{cd}\left(y\right){\nabla}_{y}^{e}{G}_{{F}_{n}}(x,y){\nabla}_{e}^{y}{G}_{{F}_{n}}(x,y)\right)\end{array}$$  
$$\begin{array}{ccc}& & +{\mathcal{K}}_{y}^{cd}\left(2{\nabla}_{x}^{a}{G}_{{F}_{n}}(x,y){\nabla}_{x}^{b}{G}_{{F}_{n}}(x,y){g}^{ab}\left(x\right){\nabla}_{x}^{e}{G}_{{F}_{n}}(x,y){\nabla}_{e}^{x}{G}_{{F}_{n}}(x,y)\right)\end{array}$$  
$$\begin{array}{ccc}& & +2{\mathcal{K}}_{x}^{ab}{\mathcal{K}}_{y}^{cd}\left({G}_{{F}_{n}}^{2}(x,y)\right)],\end{array}$$ 
(39)

where
${\mathcal{K}}_{x}^{ab}$
is the differential operator
$$\begin{array}{c}{\mathcal{K}}_{x}^{ab}\equiv \xi \left({g}^{ab}\left(x\right){\square}_{x}{\nabla}_{x}^{a}{\nabla}_{x}^{b}+{G}^{ab}\left(x\right)\right)\frac{1}{2}{m}^{2}{g}^{ab}\left(x\right).\end{array}$$ 
(40)

Note that, in the vacuum state
$0\rangle $
, the term
$\langle {\hat{\phi}}_{n}^{2}\left(x\right)\rangle $
in Equation ( 34 ) can also be written as
$\langle {\hat{\phi}}_{n}^{2}\left(x\right)\rangle =i{G}_{{F}_{n}}(x,x)={G}_{n}^{+}(x,x)$
.
Finally, the causality of the Einstein–Langevin equation (
34 ) can be explicitly seen as follows.
The nonlocal terms in that equation are due to the kernel
$H(x,y)$
which is defined in Equation ( 22 ) as the sum of
${H}_{S}(x,y)$
and
${H}_{A}(x,y)$
. Now, when the points
$x$
and
$y$
are spacelike separated,
${\hat{\phi}}_{n}\left(x\right)$
and
${\hat{\phi}}_{n}\left(y\right)$
commute and, thus,
${G}_{n}^{+}(x,y)=i{G}_{{F}_{n}}(x,y)=\frac{1}{2}\langle 0\left\right\{{\hat{\phi}}_{n}\left(x\right),{\hat{\phi}}_{n}\left(y\right)\left\}\right0\rangle $
, which is real. Hence, from the above expressions, we have that
${H}_{{A}_{n}}^{abcd}(x,y)={H}_{{S}_{n}}^{abcd}(x,y)=0$
, and thus
${H}_{n}^{abcd}(x,y)=0$
. This fact is expected since, from the causality of the expectation value of the stressenergy operator [
282]
, we know that the nonlocal dependence on the metric perturbation in the Einstein–Langevin equation, see Equation ( 14 ), must be causal. See [
169]
for an alternative proof of the causal nature of the Einstein–Langevin equation.
5 Noise Kernel and PointSeparation
In this section we explore further the properties of the noise kernel and the stressenergy bitensor. Similar to what was done for the stressenergy tensor it is desirable to relate the noise kernel defined at separated points to the Green function of a quantum field. We pointed out earlier [
154]
that field quantities defined at two separated points may possess important information which could be the starting point for probes into possible extended structures of spacetime. Of more practical concern is how one can define a finite quantity at one point or in some small region around it from the noise kernel defined at two separated points. When we refer to, say, the fluctuations of energy density in ordinary (pointwise) quantum field theory, we are in actuality asking such a question. This is essential for addressing fundamental issues like

∙
the validity of semiclassical gravity [194] – e.g., whether the fluctuations to mean ratio is a correct criterion [163, 242, 93, 95, 10, 9] ;

∙
whether the fluctuations in the vacuum energy density which drives some models of inflationary cosmology violates the positive energy condition;

∙
physical effects of black hole horizon fluctuations and Hawking radiation backreaction – to begin with, is the fluctuations finite or infinite?

∙
general relativity as a low energy effective theory in the geometrohydrodynamic limit towards a kinetic theory approach to quantum gravity [152, 154, 155] .
Thus, for comparison with ordinary phenomena at low energy we need to find a reasonable prescription for obtaining a finite quantity of the noise kernel in the limit of ordinary (pointdefined) quantum field theory. Regularization schemes used in obtaining a finite expression for the stressenergy tensor have been applied to the noise kernel^{
$\text{2}$
}
.
This includes the simple normal ordering [
194,
295]
and smeared field operator [
242]
methods applied to the Minkowski and Casimir spaces, zetafunction [
87,
189,
53]
for spacetimes with an Euclidean section, applied to the Casimir effect [
69]
and the Einstein Universe [
241]
, or the covariant pointseparation methods applied to the Minkowski [
242]
, hot flat space and the Schwarzschild spacetime [
245]
. There are differences and deliberations on whether it is meaningful to seek a pointwise expression for the noise kernel, and if so what is the correct way to proceed – e.g., regularization by a subtraction scheme or by integrating over a testfield. Intuitively the smear field method [
242]
may better preserve the integrity of the noise kernel as it provides a sampling of the two point function rather than using a subtraction scheme which alters its innate properties by forcing a nonlocal quantity into a local one. More investigation is needed to clarify these points, which bear on important issues like the validity of semiclassical gravity. We shall set a more modest goal here, to derive a general expression for the noise kernel for quantum fields in an arbitrary curved spacetime in terms of Green functions and leave the discussion of pointwise limit to a later date. For this purpose the covariant pointseparation method which highlights the bitensor features, when used not as a regularization scheme, is perhaps closest to the spirit of stochastic gravity.
The task of finding a general expression of the noisekernel for quantum fields in curved spacetimes was carried out by Phillips and Hu in two papers using the “modified” point separation
scheme [
281,
1,
283]
. Their first paper [
243]
begins with a discussion of the procedures for dealing with the quantum stress tensor bioperator at two separated points, and ends with a general expression of the noise kernel defined at separated points expressed as products of covariant derivatives up to the fourth order of the quantum field's Green function. (The stress tensor involves up to two covariant derivatives.) This result holds for
$x\ne y$
without recourse to renormalization of the Green function, showing that
${N}_{ab{c}^{\prime}{d}^{\prime}}(x,y)$
is always finite for
$x\ne y$
(and off the light cone for massless theories). In particular, for a massless conformally coupled free scalar field on a four dimensional manifold, they computed the trace of the noise kernel at both points and found this double trace vanishes identically. This implies that there is no stochastic correction to the trace anomaly for massless conformal fields, in agreement with results arrived at in [
44,
58,
208]
(see also Section 3 ). In their second paper [
245]
a Gaussian approximation for the Green function (which is what limits the accuracy of the results) is used to derive finite expressions for two specific classes of spacetimes, ultrastatic spacetimes, such as the hot flat space, and the conformallyultrastatic spacetimes, such as the Schwarzschild spacetime. Again, the validity of these results may depend on how we view the relevance and meaning of regularization. We will only report the result of their first paper here.
5.1 Point separation
The point separation scheme introduced in the 1960s by DeWitt [
74]
was brought to more popular use in the 1970s in the context of quantum field theory in curved spacetimes [
75,
67,
68]
as a means for obtaining a finite quantum stress tensor. Since the stressenergy tensor is built from the product of a pair of field operators evaluated at a single point, it is not welldefined.
In this scheme, one introduces an artificial separation of the single point
$x$
to a pair of closely separated points
$x$
and
${x}^{\prime}$
. The problematic terms involving field products such as
$\hat{\phi}(x{)}^{2}$
becomes
$\hat{\phi}\left(x\right)\hat{\phi}\left({x}^{\prime}\right)$
, whose expectation value is well defined. If one is interested in the low energy behavior captured by the pointdefined quantum field theory – as the effort in the 1970s was directed – one takes the coincidence limit. Once the divergences present are identified, they may be removed (regularization) or moved (by renormalizing the coupling constants), to produce a welldefined, finite stress tensor at a single point.
Thus the first order of business is the construction of the stress tensor and then to derive the symmetric stressenergy tensor two point function, the noise kernel, in terms of the Wightman Green function. In this section we will use the traditional notation for index tensors in the pointseparation context.
5.1.1 ntensors and endpoint expansions
An object like the Green function
$G(x,y)$
is an example of a biscalar: It transforms as scalar at both points
$x$
and
$y$
. We can also define a bitensor
${T}_{{a}_{1}...{a}_{n}{b}_{1}^{\prime}...{b}_{m}^{\prime}}(x,y)$
: Upon a coordinate transformation, this transforms as a rank
$n$
tensor at
$x$
and a rank
$m$
tensor at
$y$
. We will extend this up to a quadtensor
${T}_{{a}_{1}...{a}_{{n}_{1}}{b}_{1}^{\prime}...{b}_{{n}_{2}}^{\prime}{c}_{1}^{\prime \prime}...{c}_{{n}_{3}}^{\prime \prime}{d}_{1}^{\prime \prime \prime}...{d}_{{n}_{4}}^{\prime \prime \prime}}$
which has support at four points
$x,y,{x}^{\prime},{y}^{\prime}$
, transforming as rank
${n}_{1},{n}_{2},{n}_{3},{n}_{4}$
tensors at each of the four points. This also sets the notation we will use: unprimed indices referring to the tangent space constructed above
$x$
, single primed indices to
$y$
, double primed to
${x}^{\prime}$
and triple primed to
${y}^{\prime}$
. For each point, there is the covariant derivative
${\nabla}_{a}$
at that point. Covariant derivatives at different points commute, and the covariant derivative at, say, point
${x}^{\prime}$
does not act on a bitensor defined at, say,
$x$
and
$y$
:
$$\begin{array}{c}{T}_{a{b}^{\prime};c;{d}^{\prime}}={T}_{a{b}^{\prime};{d}^{\prime};c}\text{and}{T}_{a{b}^{\prime};{c}^{\prime \prime}}=0.\end{array}$$ 
(41)

To simplify notation, henceforth we will eliminate the semicolons after the first one for multiple covariant derivatives at multiple points.
Having objects defined at different points, the
coincident limit is defined as evaluation “on the diagonal”, in the sense of the spacetime support of the function or tensor, and the usual shorthand
$\left[G(x,y)\right]\equiv G(x,x)$
is used. This extends to
$n$
tensors as
$$\begin{array}{c}\left[{T}_{{a}_{1}...{a}_{{n}_{1}}{b}_{1}^{\prime}...{b}_{{n}_{2}}^{\prime}{c}_{1}^{\prime \prime}...{c}_{{n}_{3}}^{\prime \prime}{d}_{1}^{\prime \prime \prime}...{d}_{{n}_{4}}^{\prime \prime \prime}}\right]={T}_{{a}_{1}...{a}_{{n}_{1}}{b}_{1}...{b}_{{n}_{2}}{c}_{1}...{c}_{{n}_{3}}{d}_{1}...{d}_{{n}_{4}}},\end{array}$$ 
(42)

i.e., this becomes a rank
$({n}_{1}+{n}_{2}+{n}_{3}+{n}_{4})$
tensor at
$x$
. The multivariable chain rule relates covariant derivatives acting at different points, when we are interested in the coincident limit:
$$\begin{array}{c}{\left[{T}_{{a}_{1}...{a}_{m}{b}_{1}^{\prime}...{b}_{n}^{\prime}}\right]}_{;c}=\left[{T}_{{a}_{1}...{a}_{m}{b}_{1}^{\prime}...{b}_{n}^{\prime};c}\right]+\left[{T}_{{a}_{1}...{a}_{m}{b}_{1}^{\prime}...{b}_{n}^{\prime};{c}^{\prime}}\right].\end{array}$$ 
(43)

This result is referred to as Synge's theorem in this context; we follow Fulling's discussion [
100]
.
The bitensor of
parallel transport
${{g}_{a}}^{{b}^{\prime}}$
is defined such that when it acts on a vector
${v}_{{b}^{\prime}}$
at
$y$
, it parallel transports the vector along the geodesics connecting
$x$
and
$y$
. This allows us to add vectors and tensors defined at different points. We cannot directly add a vector
${v}_{a}$
at
$x$
and vector
${w}_{{a}^{\prime}}$
at
$y$
. But by using
${{g}_{a}}^{{b}^{\prime}}$
, we can construct the sum
${v}^{a}+{{g}_{a}}^{{b}^{\prime}}{w}_{{b}^{\prime}}$
. We will also need the obvious property
$\left[{{g}_{a}}^{{b}^{\prime}}\right]={{g}_{a}}^{b}$
.
The main biscalar we need is the
world function
$\sigma (x,y)$
. This is defined as a half of the square of the geodesic distance between the points
$x$
and
$y$
. It satisfies the equation
$$\begin{array}{c}\sigma =\frac{1}{2}{\sigma}^{;p}{\sigma}_{;p}.\end{array}$$ 
(44)

Often in the literature, a covariant derivative is implied when the world function appears with indices,
${\sigma}^{a}\equiv {\sigma}^{;a}$
, i.e., taking the covariant derivative at
$x$
, while
${\sigma}^{{a}^{\prime}}$
means the covariant derivative at
$y$
. This is done since the vector
${\sigma}^{a}$
is the tangent vector to the geodesic with length equal to the distance between
$x$
and
$y$
. As
${\sigma}^{a}$
records information about distance and direction for the two points, this makes it ideal for constructing a series expansion of a biscalar. The end point expansion of a biscalar
$S(x,y)$
is of the form
$$\begin{array}{c}S(x,y)={A}^{\left(0\right)}+{\sigma}^{p}{A}_{p}^{\left(1\right)}+{\sigma}^{p}{\sigma}^{q}{A}_{pq}^{\left(2\right)}+{\sigma}^{p}{\sigma}^{q}{\sigma}^{r}{A}_{pqr}^{\left(3\right)}+{\sigma}^{p}{\sigma}^{q}{\sigma}^{r}{\sigma}^{s}{A}_{pqrs}^{\left(4\right)}+...,\end{array}$$ 
(45)

where, following our convention, the expansion tensors
${A}_{{a}_{1}...{a}_{n}}^{\left(n\right)}$
with unprimed indices have support at
$x$
(hence the name end point expansion). Only the symmetric part of these tensors contribute to the expansion. For the purposes of multiplying series expansions it is convenient to separate the distance dependence from the direction dependence. This is done by introducing the unit vector
${p}^{a}={\sigma}^{a}/\sqrt{2\sigma}$
. Then the series expansion can be written
$$\begin{array}{c}S(x,y)={A}^{\left(0\right)}+{\sigma}^{\frac{1}{2}}{A}^{\left(1\right)}+\sigma {A}^{\left(2\right)}+{\sigma}^{\frac{3}{2}}{A}^{\left(3\right)}+{\sigma}^{2}{A}^{\left(4\right)}+....\end{array}$$ 
(46)

The expansion scalars are related, via
${A}^{\left(n\right)}={2}^{n/2}{A}_{{p}_{1}...{p}_{n}}^{\left(n\right)}{p}^{{p}_{1}}...{p}^{{p}_{n}}$
, to the expansion tensors.
The last object we need is the
VanVleck–Morette determinant
$D(x,y)$
, defined as
$D(x,y)\equiv det({\sigma}_{;a{b}^{\prime}})$
. The related biscalar
$$\begin{array}{c}{\Delta}^{1/2}={\left(\frac{D(x,y)}{\sqrt{g\left(x\right)g\left(y\right)}}\right)}^{\frac{1}{2}}\end{array}$$ 
(47)

satisfies the equation
$$\begin{array}{c}{\Delta}^{1/2}\left(4{{\sigma}_{;p}}^{p}\right)2{\Delta}_{;p}^{1/2}{\sigma}^{;p}=0\end{array}$$ 
(48)

with the boundary condition
$\left[{\Delta}^{1/2}\right]=1$
.
Further details on these objects and discussions of the definitions and properties are contained in [
67,
68]
and [
240]
. There it is shown how the defining equations for
$\sigma $
and
${\Delta}^{1/2}$
are used to determine the coincident limit expression for the various covariant derivatives of the world function (
$\left[{\sigma}_{;a}\right]$
,
$\left[{\sigma}_{;ab}\right]$
, etc.) and how the defining differential equation for
${\Delta}^{1/2}$
can be used to determine the series expansion of
${\Delta}^{1/2}$
. We show how the expansion tensors
${A}_{{a}_{1}...{a}_{n}}^{\left(n\right)}$
are determined in terms of the coincident limits of covariant derivatives of the biscalar
$S(x,y)$
. ([
240]
details how point separation can be implemented on the computer to provide easy access to a wider range of applications involving higher derivatives of the curvature tensors.)
5.2 Stressenergy bitensor operator and noise kernel
Even though we believe that the pointseparated results are more basic in the sense that it reflects a deeper structure of the quantum theory of spacetime, we will nevertheless start with quantities defined at one point, because they are what enter in conventional quantum field theory.
We will use point separation to introduce the biquantities. The key issue here is thus the distinction between pointdefined (
pt) and pointseparated (bi) quantities.
For a free classical scalar field
$\phi $
with the action
${S}_{m}[g,\phi ]$
defined in Equation ( 1 ), the classical stressenergy tensor is
$$\begin{array}{ccc}{T}_{ab}& =& (12\xi ){\phi}_{;a}{\phi}_{;b}+\left(2\xi \frac{1}{2}\right){\phi}_{;p}{\phi}^{;p}{g}_{ab}+2\xi \phi \left({{\phi}_{;p}}^{p}{{\phi}_{;a}}_{b}{g}_{ab}\right)\end{array}$$  
$$\begin{array}{ccc}& & +{\phi}^{2}\xi \left({R}_{ab}\frac{1}{2}R{g}_{ab}\right)\frac{1}{2}{m}^{2}{\phi}^{2}{g}_{ab},\end{array}$$ 
(49)

which is equivalent to the tensor of Equation ( 3 ), but written in a slightly different form for convenience. When we make the transition to quantum field theory, we promote the field
$\phi \left(x\right)$
to a field operator
$\hat{\phi}\left(x\right)$
. The fundamental problem of defining a quantum operator for the stress tensor is immediately visible: The field operator appears quadratically. Since
$\hat{\phi}\left(x\right)$
is an operatorvalued distribution, products at a single point are not welldefined. But if the product is point separated,
${\hat{\phi}}^{2}\left(x\right)\to \hat{\phi}\left(x\right)\hat{\phi}\left({x}^{\prime}\right)$
, they are finite and welldefined.
Let us first seek a pointseparated extension of these classical quantities and then consider the quantum field operators. Point separation is symmetrically extended to products of covariant derivatives of the field according to
$$\begin{array}{ccc}\left({\phi}_{;a}\right)\left({\phi}_{;b}\right)& \to & \frac{1}{2}\left({{g}_{a}}^{{p}^{\prime}}{\nabla}_{{p}^{\prime}}{\nabla}_{b}+{{g}_{b}}^{{p}^{\prime}}{\nabla}_{a}{\nabla}_{{p}^{\prime}}\right)\phi \left(x\right)\phi \left({x}^{\prime}\right),\end{array}$$  
$$\begin{array}{ccc}\phi \left({\phi}_{;ab}\right)& \to & \frac{1}{2}\left({\nabla}_{a}{\nabla}_{b}+{{g}_{a}}^{{p}^{\prime}}{{g}_{b}}^{{q}^{\prime}}{\nabla}_{{p}^{\prime}}{\nabla}_{{q}^{\prime}}\right)\phi \left(x\right)\phi \left({x}^{\prime}\right).\end{array}$$  
The bivector of parallel displacement
${{g}_{a}}^{{a}^{\prime}}(x,{x}^{\prime})$
is included so that we may have objects that are rank 2 tensors at
$x$
and scalars at
${x}^{\prime}$
.
To carry out point separation on Equation (
49 ), we first define the differential operator
$$\begin{array}{ccc}{\mathcal{T}}_{ab}& =& \frac{1}{2}\left(12\xi \right)\left({{g}_{a}}^{{a}^{\prime}}{\nabla}_{{a}^{\prime}}{\nabla}_{b}+{{g}_{b}}^{{b}^{\prime}}{\nabla}_{a}{\nabla}_{{b}^{\prime}}\right)+\left(2\xi \frac{1}{2}\right){g}_{ab}{g}^{c{d}^{\prime}}{\nabla}_{c}{\nabla}_{{d}^{\prime}}\end{array}$$  
$$\begin{array}{ccc}& & \xi \left({\nabla}_{a}{\nabla}_{b}+{{g}_{a}}^{{a}^{\prime}}{{g}_{b}}^{{b}^{\prime}}{\nabla}_{{a}^{\prime}}{\nabla}_{{b}^{\prime}}\right)+\xi {g}_{ab}\left({\nabla}_{c}{\nabla}^{c}+{\nabla}_{{c}^{\prime}}{\nabla}^{{c}^{\prime}}\right)+\xi \left({R}_{ab}\frac{1}{2}{g}_{ab}R\right)\frac{1}{2}{m}^{2}{g}_{ab},\end{array}$$ 
(50)

from which we obtain the classical stress tensor as
$$\begin{array}{c}{T}_{ab}\left(x\right)={lim}_{{x}^{\prime}\to x}{\mathcal{T}}_{ab}\phi \left(x\right)\phi \left({x}^{\prime}\right).\end{array}$$ 
(51)

That the classical tensor field no longer appears as a product of scalar fields at a single point allows a smooth transition to the quantum tensor field. From the viewpoint of the stress tensor, the separation of points is an artificial construct, so when promoting the classical field to a quantum one, neither point should be favored. The product of field configurations is taken to be the symmetrized operator product, denoted by curly brackets:
$$\begin{array}{c}\phi \left(x\right)\phi \left(y\right)\to \frac{1}{2}\left\{\hat{\phi}\left(x\right),\hat{\phi}\left(y\right)\right\}=\frac{1}{2}\left(\hat{\phi}\left(x\right)\hat{\phi}\left(y\right)+\hat{\phi}\left(y\right)\hat{\phi}\left(x\right)\right)\end{array}$$ 
(52)

With this, the point separated stressenergy tensor operator is defined as
$$\begin{array}{c}{\hat{T}}_{ab}(x,{x}^{\prime})\equiv \frac{1}{2}{\mathcal{T}}_{ab}\left\{\hat{\phi}\left(x\right),\hat{\phi}\left({x}^{\prime}\right)\right\}.\end{array}$$ 
(53)

While the classical stress tensor was defined at the coincidence limit
${x}^{\prime}\to x$
, we cannot attach any physical meaning to the quantum stress tensor at one point until the issue of regularization is dealt with, which will happen in the next section. For now, we will maintain point separation so as to have a mathematically meaningful operator.
The expectation value of the pointseparated stress tensor can now be taken. This amounts to replacing the field operators by their expectation value, which is given by the Hadamard (or Schwinger) function
$$\begin{array}{c}{G}^{\left(1\right)}(x,{x}^{\prime})=\langle \left\{\hat{\phi}\left(x\right),\hat{\phi}\left({x}^{\prime}\right)\right\}\rangle ,\end{array}$$ 
(54)

and the pointseparated stress tensor is defined as
$$\begin{array}{c}\langle {\hat{T}}_{ab}(x,{x}^{\prime})\rangle =\frac{1}{2}{\mathcal{T}}_{ab}{G}^{\left(1\right)}(x,{x}^{\prime}),\end{array}$$ 
(55)

where, since
${\mathcal{T}}_{ab}$
is a differential operator, it can be taken “outside” the expectation value.
The expectation value of the pointseparated quantum stress tensor for a free, massless (
$m=0$
) conformally coupled (
$\xi =\frac{1}{6}$
) scalar field on a four dimension spacetime with scalar curvature
$R$
is
$$\begin{array}{ccc}\langle {\hat{T}}_{ab}(x,{x}^{\prime})\rangle & =& \frac{1}{6}\left({g}_{b}^{{p}^{\prime}}{G}_{;{p}^{\prime}a}^{\left(1\right)}+{g}_{a}^{{p}^{\prime}}{G}_{;{p}^{\prime}b}^{\left(1\right)}\right)\frac{1}{12}{g}_{q}^{{p}^{\prime}}{{G}_{;{p}^{\prime}}^{\left(1\right)}}^{q}{g}_{ab}\end{array}$$  
$$\begin{array}{ccc}& & \frac{1}{12}\left({g}_{a}^{{p}^{\prime}}{g}_{b}^{{q}^{\prime}}{G}_{;{p}^{\prime}{q}^{\prime}}^{\left(1\right)}+{G}_{;ab}^{\left(1\right)}\right)+\frac{1}{12}\left[\left({{G}_{;{p}^{\prime}}^{\left(1\right)}}^{{p}^{\prime}}+{{G}_{;p}^{\left(1\right)}}^{p}\right){g}_{ab}\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{12}{G}^{\left(1\right)}\left({R}_{ab}\frac{1}{2}R{g}_{ab}\right).\end{array}$$ 
(56)

5.2.1 Finiteness of the noise kernel
We now turn our attention to the noise kernel introduced in Equation ( 11 ), which is the symmetrized product of the (mean subtracted) stress tensor operator:
$$\begin{array}{ccc}8{N}_{ab,{c}^{\prime}{d}^{\prime}}(x,y)& =& \langle \left\{{\hat{T}}_{ab}\left(x\right)\langle {\hat{T}}_{ab}\left(x\right)\rangle ,{\hat{T}}_{{c}^{\prime}{d}^{\prime}}\left(y\right)\langle {\hat{T}}_{{c}^{\prime}{d}^{\prime}}\left(y\right)\rangle \right\}\rangle \end{array}$$  
$$\begin{array}{ccc}& =& \langle \left\{{\hat{T}}_{ab}\left(x\right),{\hat{T}}_{{c}^{\prime}{d}^{\prime}}\left(y\right)\right\}\rangle 2\langle {\hat{T}}_{ab}\left(x\right)\rangle \langle {\hat{T}}_{{c}^{\prime}{d}^{\prime}}\left(y\right)\rangle .\end{array}$$ 
(57)

Since
${\hat{T}}_{ab}\left(x\right)$
defined at one point can be illbehaved as it is generally divergent, one can question the soundness of these quantities. But as will be shown later, the noise kernel is finite for
$y\ne x$
. All field operator products present in the first expectation value that could be divergent, are canceled by similar products in the second term. We will replace each of the stress tensor operators in the above expression for the noise kernel by their point separated versions, effectively separating the two points
$(x,y)$
into the four points
$(x,{x}^{\prime},y,{y}^{\prime})$
. This will allow us to express the noise kernel in terms of a pair of differential operators acting on a combination of four and two point functions. Wick's theorem will allow the four point functions to be reexpressed in terms of two point functions. From this we see that all possible divergences for
$y\ne x$
will cancel. When the coincidence limit is taken, divergences do occur. The above procedure will allow us to isolate the divergences and to obtain a finite result.
Taking the pointseparated quantities as more basic, one should replace each of the stress tensor operators in the above with the corresponding point separated version (
53 ), with
${\mathcal{T}}_{ab}$
acting at
$x$
and
${x}^{\prime}$
and
${\mathcal{T}}_{{c}^{\prime}{d}^{\prime}}$
acting at
$y$
and
${y}^{\prime}$
. In this framework the noise kernel is defined as
$$\begin{array}{c}8{N}_{ab,{c}^{\prime}{d}^{\prime}}(x,y)={lim}_{{x}^{\prime}\to x}{lim}_{{y}^{\prime}\to y}{\mathcal{T}}_{ab}{\mathcal{T}}_{{c}^{\prime}{d}^{\prime}}G(x,{x}^{\prime},y,{y}^{\prime}),\end{array}$$ 
(58)

where the four point function is
$$\begin{array}{c}G(x,{x}^{\prime},y,{y}^{\prime})=\frac{1}{4}\left[\langle \left\{\left\{\hat{\phi}\left(x\right),\hat{\phi}\left({x}^{\prime}\right)\right\},\left\{\hat{\phi}\left(y\right),\hat{\phi}\left({y}^{\prime}\right)\right\}\right\}\rangle 2\langle \left\{\hat{\phi}\left(x\right),\hat{\phi}\left({x}^{\prime}\right)\right\}\rangle \langle \left\{\hat{\phi}\left(y\right),\hat{\phi}\left({y}^{\prime}\right)\right\}\rangle \right].\end{array}$$ 
(59)

We assume that the pairs
$(x,{x}^{\prime})$
and
$(y,{y}^{\prime})$
are each within their respective Riemann normal coordinate neighborhoods so as to avoid problems that possible geodesic caustics might be present.
When we later turn our attention to computing the limit
$y\to x$
, after issues of regularization are addressed, we will want to assume that all four points are within the same Riemann normal coordinate neighborhood.
Wick's theorem, for the case of free fields which we are considering, gives the simple product four point function in terms of a sum of products of Wightman functions (we use the shorthand notation
${G}_{xy}\equiv {G}_{+}(x,y)=\langle \hat{\phi}\left(x\right)\hat{\phi}\left(y\right)\rangle $
):
$$\begin{array}{c}\langle \hat{\phi}\left(x\right)\hat{\phi}\left(y\right)\hat{\phi}\left({x}^{\prime}\right)\hat{\phi}\left({y}^{\prime}\right)\rangle ={G}_{x{y}^{\prime}}{G}_{y{x}^{\prime}}+{G}_{x{x}^{\prime}}{G}_{y{y}^{\prime}}+{G}_{xy}{G}_{{x}^{\prime}{y}^{\prime}}\end{array}$$ 
(60)

Expanding out the anticommutators in Equation ( 59 ) and applying Wick's theorem, the four point function becomes
$$\begin{array}{c}G(x,{x}^{\prime},y,{y}^{\prime})={G}_{x{y}^{\prime}}{G}_{{x}^{\prime}y}+{G}_{xy}{G}_{{x}^{\prime}{y}^{\prime}}+{G}_{y{x}^{\prime}}{G}_{{y}^{\prime}x}+{G}_{yx}{G}_{{y}^{\prime}{x}^{\prime}}.\end{array}$$ 
(61)

We can now easily see that the noise kernel defined via this function is indeed well defined for the limit
$({x}^{\prime},{y}^{\prime})\to (x,y)$
:
$$\begin{array}{c}G(x,x,y,y)=2\left({G}_{xy}^{2}+{G}_{yx}^{2}\right).\end{array}$$ 
(62)

From this we can see that the noise kernel is also well defined for
$y\ne x$
; any divergence present in the first expectation value of Equation ( 59 ) have been cancelled by those present in the pair of Green functions in the second term, in agreement with the results of Section 3 .
5.2.2 Explicit form of the noise kernel
We will let the points separated for a while so we can keep track of which covariant derivative acts on which arguments of which Wightman function. As an example (the complete calculation is quite long), consider the result of the first set of covariant derivative operators in the differential operator ( 50 ), from both
${\mathcal{T}}_{ab}$
and
${\mathcal{T}}_{{c}^{\prime}{d}^{\prime}}$
, acting on
$G(x,{x}^{\prime},y,{y}^{\prime})$
:
$$\begin{array}{c}\frac{1}{4}{\left(12\xi \right)}^{2}\left({{g}_{a}}^{{p}^{\prime \prime}}{\nabla}_{{p}^{\prime \prime}}{\nabla}_{b}+{{g}_{b}}^{{p}^{\prime \prime}}{\nabla}_{{p}^{\prime \prime}}{\nabla}_{a}\right)\left({{g}_{{c}^{\prime}}}^{{q}^{\prime \prime \prime}}{\nabla}_{{q}^{\prime \prime \prime}}{\nabla}_{{d}^{\prime}}+{{g}_{{d}^{\prime}}}^{{q}^{\prime \prime \prime}}{\nabla}_{{q}^{\prime \prime \prime}}{\nabla}_{{c}^{\prime}}\right)G(x,{x}^{\prime},y,{y}^{\prime}).\end{array}$$ 
(63)

(Our notation is that
${\nabla}_{a}$
acts at
$x$
,
${\nabla}_{{c}^{\prime}}$
at
$y$
,
${\nabla}_{{b}^{\prime \prime}}$
at
${x}^{\prime}$
, and
${\nabla}_{{d}^{\prime \prime \prime}}$
at
${y}^{\prime}$
.) Expanding out the differential operator above, we can determine which derivatives act on which Wightman function:
$$\begin{array}{ccc}\frac{{\left(12\xi \right)}^{2}}{4}& \times & [{{g}_{{c}^{\prime}}}^{{p}^{\prime \prime \prime}}{g}_{a}^{{q}^{\prime \prime}}({G}_{x{y}^{\prime};b{p}^{\prime \prime \prime}}{G}_{{x}^{\prime}y;{q}^{\prime \prime}{d}^{\prime}}+{G}_{xy;b{d}^{\prime}}{G}_{{x}^{\prime}{y}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}}\end{array}$$  
$$\begin{array}{ccc}& & +{G}_{y{x}^{\prime};{q}^{\prime \prime}{d}^{\prime}}{G}_{{y}^{\prime}x;b{p}^{\prime \prime \prime}}+{G}_{yx;b{d}^{\prime}}{G}_{{y}^{\prime}{x}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}})\end{array}$$  
$$\begin{array}{ccc}& & +{{g}_{{d}^{\prime}}}^{{p}^{\prime \prime \prime}}{g}_{a}^{{q}^{\prime \prime}}({G}_{x{y}^{\prime};b{p}^{\prime \prime \prime}}{G}_{{x}^{\prime}y;{q}^{\prime \prime}{c}^{\prime}}+{G}_{xy;b{c}^{\prime}}{G}_{{x}^{\prime}{y}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}}\end{array}$$  
$$\begin{array}{ccc}& & +{G}_{y{x}^{\prime};{q}^{\prime \prime}{c}^{\prime}}{G}_{{y}^{\prime}x;b{p}^{\prime \prime \prime}}+{G}_{yx;b{c}^{\prime}}{G}_{{y}^{\prime}{x}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}})\end{array}$$  
$$\begin{array}{ccc}& & +{{g}_{{c}^{\prime}}}^{{p}^{\prime \prime \prime}}{g}_{b}^{{q}^{\prime \prime}}({G}_{x{y}^{\prime};a{p}^{\prime \prime \prime}}{G}_{{x}^{\prime}y;{q}^{\prime \prime}{d}^{\prime}}+{G}_{xy;a{d}^{\prime}}{G}_{{x}^{\prime}{y}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}}\end{array}$$  
$$\begin{array}{ccc}& & +{G}_{y{x}^{\prime};{q}^{\prime \prime}{d}^{\prime}}{G}_{{y}^{\prime}x;a{p}^{\prime \prime \prime}}+{G}_{yx;a{d}^{\prime}}{G}_{{y}^{\prime}{x}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}})\end{array}$$  
$$\begin{array}{ccc}& & +{{g}_{{d}^{\prime}}}^{{p}^{\prime \prime \prime}}{g}_{b}^{{q}^{\prime \prime}}({G}_{x{y}^{\prime};a{p}^{\prime \prime \prime}}{G}_{{x}^{\prime}y;{q}^{\prime \prime}{c}^{\prime}}+{G}_{xy;a{c}^{\prime}}{G}_{{x}^{\prime}{y}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}}\end{array}$$  
$$\begin{array}{ccc}& & +{G}_{y{x}^{\prime};{q}^{\prime \prime}{c}^{\prime}}{G}_{{y}^{\prime}x;a{p}^{\prime \prime \prime}}+{G}_{yx;a{c}^{\prime}}{G}_{{y}^{\prime}{x}^{\prime};{q}^{\prime \prime}{p}^{\prime \prime \prime}})].\end{array}$$ 
(64)

If we now let
${x}^{\prime}\to x$
and
${y}^{\prime}\to y$
, the contribution to the noise kernel is (including the factor of
$\frac{1}{8}$
present in the definition of the noise kernel):
$$\begin{array}{c}\frac{1}{8}\left[{\left(12\xi \right)}^{2}\left({G}_{xy;a{d}^{\prime}}{G}_{xy;b{c}^{\prime}}+{G}_{xy;a{c}^{\prime}}{G}_{xy;b{d}^{\prime}}\right)+{\left(12\xi \right)}^{2}\left({G}_{yx;a{d}^{\prime}}{G}_{yx;b{c}^{\prime}}+{G}_{yx;a{c}^{\prime}}{G}_{yx;b{d}^{\prime}}\right)\right].\end{array}$$ 
(65)

That this term can be written as the sum of a part involving
${G}_{xy}$
and one involving
${G}_{yx}$
is a general property of the entire noise kernel. It thus takes the form
$$\begin{array}{c}{N}_{ab{c}^{\prime}{d}^{\prime}}(x,y)={N}_{ab{c}^{\prime}{d}^{\prime}}\left[{G}_{+}(x,y)\right]+{N}_{ab{c}^{\prime}{d}^{\prime}}\left[{G}_{+}(y,x)\right].\end{array}$$ 
(66)

We will present the form of the functional
${N}_{ab{c}^{\prime}{d}^{\prime}}\left[G\right]$
shortly. First we note, that for
$x$
and
$y$
timelike separated, the above split of the noise kernel allows us to express it in terms of the Feynman (time ordered) Green function
${G}_{F}(x,y)$
and the Dyson (antitime ordered) Green function
${G}_{D}(x,y)$
:
$$\begin{array}{c}{N}_{ab{c}^{\prime}{d}^{\prime}}(x,y)={N}_{ab{c}^{\prime}{d}^{\prime}}\left[{G}_{F}(x,y)\right]+{N}_{ab{c}^{\prime}{d}^{\prime}}\left[{G}_{D}(x,y)\right].\end{array}$$ 
(67)

This can be connected with the zeta function approach to this problem [
241]
as follows: Recall when the quantum stress tensor fluctuations determined in the Euclidean section is analytically continued back to Lorentzian signature (
$\tau \to it$
), the time ordered product results. On the other hand, if the continuation is
$\tau \to it$
, the antitime ordered product results. With this in mind, the noise kernel is seen to be related to the quantum stress tensor fluctuations derived via the effective action as
$$\begin{array}{c}16{N}_{ab{c}^{\prime}{d}^{\prime}}={\Delta {T}_{ab{c}^{\prime}{d}^{\prime}}^{2}}_{t=i\tau ,{t}^{\prime}=i{\tau}^{\prime}}+{\Delta {T}_{ab{c}^{\prime}{d}^{\prime}}^{2}}_{t=i\tau ,{t}^{\prime}=i{\tau}^{\prime}}.\end{array}$$ 
(68)

The complete form of the functional
${N}_{ab{c}^{\prime}{d}^{\prime}}\left[G\right]$
is
$$\begin{array}{c}{N}_{ab{c}^{\prime}{d}^{\prime}}\left[G\right]={\stackrel{~}{N}}_{ab{c}^{\prime}{d}^{\prime}}\left[G\right]+{g}_{ab}{\stackrel{~}{N}}_{{c}^{\prime}{d}^{\prime}}\left[G\right]+{g}_{{c}^{\prime}{d}^{\prime}}{\stackrel{~}{N}}_{ab}^{\prime}\left[G\right]+{g}_{ab}{g}_{{c}^{\prime}{d}^{\prime}}\stackrel{~}{N}\left[G\right],\end{array}$$ 
(69)

with
$$\begin{array}{ccc}8{\stackrel{~}{N}}_{ab{c}^{\prime}{d}^{\prime}}\left[G\right]& =& {\left(12\xi \right)}^{2}\left({G}_{;{c}^{\prime}b}{G}_{;{d}^{\prime}a}+{G}_{;{c}^{\prime}a}{G}_{;{d}^{\prime}b}\right)+4{\xi}^{2}\left({G}_{;{c}^{\prime}{d}^{\prime}}{G}_{;ab}+G{G}_{;ab{c}^{\prime}{d}^{\prime}}\right)\end{array}$$  
$$\begin{array}{ccc}& & 2\xi \left(12\xi \right)\left({G}_{;b}{G}_{;{c}^{\prime}a{d}^{\prime}}+{G}_{;a}{G}_{;{c}^{\prime}b{d}^{\prime}}+{G}_{;{d}^{\prime}}{G}_{;ab{c}^{\prime}}+{G}_{;{c}^{\prime}}{G}_{;ab{d}^{\prime}}\right)\end{array}$$  
$$\begin{array}{ccc}& & +2\xi \left(12\xi \right)\left({G}_{;a}{G}_{;b}{R}_{{c}^{\prime}{d}^{\prime}}+{G}_{;{c}^{\prime}}{G}_{;{d}^{\prime}}{R}_{ab}\right)\end{array}$$  
$$\begin{array}{ccc}& & 4{\xi}^{2}\left({G}_{;ab}{R}_{{c}^{\prime}{d}^{\prime}}+{G}_{;{c}^{\prime}{d}^{\prime}}{R}_{ab}\right)G+2{\xi}^{2}{R}_{{c}^{\prime}{d}^{\prime}}{R}_{ab}{G}^{2},\end{array}$$ 
(70)

$$\begin{array}{ccc}8{\stackrel{~}{N}}_{ab}^{\prime}\left[G\right]& =& 2(12\xi )\left[\left(2\xi \frac{1}{2}\right){G}_{;{p}^{\prime}b}{{G}_{;}}_{a}^{{p}^{\prime}}+\xi \left({G}_{;b}{{G}_{;{p}^{\prime}a}}^{{p}^{\prime}}+{G}_{;a}{{G}_{;{p}^{\prime}b}}^{{p}^{\prime}}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & 4\xi \left[\left(2\xi \frac{1}{2}\right){{G}_{;}}^{{p}^{\prime}}{G}_{;ab{p}^{\prime}}+\xi \left({{G}_{;{p}^{\prime}}}^{{p}^{\prime}}{G}_{;ab}+G{{G}_{;ab{p}^{\prime}}}^{{p}^{\prime}}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & ({m}^{2}+\xi {R}^{\prime})\left[(12\xi ){G}_{;a}{G}_{;b}2G\xi {G}_{;ab}\right]\end{array}$$  
$$\begin{array}{ccc}& & +2\xi \left[\left(2\xi \frac{1}{2}\right){G}_{;{p}^{\prime}}{{G}_{;}}^{{p}^{\prime}}+2G\xi {{G}_{;{p}^{\prime}}}^{{p}^{\prime}}\right]{R}_{ab}({m}^{2}+\xi {R}^{\prime})\xi {R}_{ab}{G}^{2},\end{array}$$ 
(71)

$$\begin{array}{ccc}8\stackrel{~}{N}\left[G\right]& =& 2{\left(2\xi \frac{1}{2}\right)}^{2}{G}_{;{p}^{\prime}q}{{{G}_{;}}^{{p}^{\prime}}}^{q}+4{\xi}^{2}\left({{G}_{;{p}^{\prime}}}^{{p}^{\prime}}{{G}_{;q}}^{q}+G{{{G}_{;p}}_{{q}^{\prime}}^{p}}^{{q}^{\prime}}\right)\end{array}$$  
$$\begin{array}{ccc}& & +4\xi \left(2\xi \frac{1}{2}\right)\left({G}_{;p}{{{G}_{;{q}^{\prime}}}^{p}}^{{q}^{\prime}}+{{G}_{;}}^{{p}^{\prime}}{{G}_{;q}}_{{p}^{\prime}}^{q}\right)\end{array}$$  
$$\begin{array}{ccc}& & \left(2\xi \frac{1}{2}\right)\left[\left({m}^{2}+\xi R\right){G}_{;{p}^{\prime}}{{G}_{;}}^{{p}^{\prime}}+\left({m}^{2}+\xi {R}^{\prime}\right){G}_{;p}{{G}^{;}}^{p}\right]\end{array}$$  
$$\begin{array}{ccc}& & 2\xi \left[\left({m}^{2}+\xi R\right){{G}_{;{p}^{\prime}}}^{{p}^{\prime}}+\left({m}^{2}+\xi {R}^{\prime}\right){{G}_{;p}}^{p}\right]G\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{2}\left({m}^{2}+\xi R\right)\left({m}^{2}+\xi {R}^{\prime}\right){G}^{2}.\end{array}$$ 
(72)

5.2.3 Trace of the noise kernel
One of the most interesting and surprising results to come out of the investigations of the quantum stress tensor undertaken in the 1970s was the discovery of the trace anomaly [
61,
84]
.
When the trace of the stress tensor
$T={g}^{ab}{T}_{ab}$
is evaluated for a field configuration that satisties the field equation ( 2 ), the trace is seen to vanish for massless conformally coupled fields. When this analysis is carried over to the renormalized expectation value of the quantum stress tensor, the trace no longer vanishes. Wald [
283]
showed that this was due to the failure of the renormalized Hadamard function
${G}_{ren}(x,{x}^{\prime})$
to be symmetric in
$x$
and
${x}^{\prime}$
, implying that it does not necessarily satisfy the field equation ( 2 ) in the variable
${x}^{\prime}$
. (The definition of
${G}_{ren}(x,{x}^{\prime})$
in the context of point separation will come next.) With this in mind, we can now determine the noise associated with the trace. Taking the trace at both points
$x$
and
$y$
of the noise kernel functional ( 67 ) yields
$$\begin{array}{ccc}N\left[G\right]& =& {g}^{ab}{g}^{{c}^{\prime}{d}^{\prime}}{N}_{ab{c}^{\prime}{d}^{\prime}}\left[G\right]\end{array}$$  
$$\begin{array}{ccc}& =& 3G\xi \left[\left({m}^{2}+\frac{1}{2}\xi R\right){{G}_{;{p}^{\prime}}}^{{p}^{\prime}}+\left({m}^{2}+\frac{1}{2}\xi {R}^{\prime}\right){{G}_{;p}}^{p}\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{9{\xi}^{2}}{2}\left({{G}_{;{p}^{\prime}}}^{{p}^{\prime}}{{G}_{;p}}^{p}+G{{{G}_{;p}}_{{p}^{\prime}}^{p}}^{{p}^{\prime}}\right)+\left({m}^{2}+\frac{1}{2}\xi R\right)\left({m}^{2}+\frac{1}{2}\xi {R}^{\prime}\right){G}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +3\left(\frac{1}{6}\xi \right)[3\left(\frac{1}{6}\xi \right){G}_{;{p}^{\prime}p}{{G}_{;}}^{{p}^{\prime}p}3\xi \left({G}_{;p}{{G}_{;{p}^{\prime}}}^{p{p}^{\prime}}+{G}_{;{p}^{\prime}}{{G}_{;p}}^{p{p}^{\prime}}\right)\end{array}$$  
$$\begin{array}{ccc}& & +\left({m}^{2}+\frac{1}{2}\xi R\right){G}_{;{p}^{\prime}}{{G}_{;}}^{{p}^{\prime}}+\left({m}^{2}+\frac{1}{2}\xi {R}^{\prime}\right){G}_{;p}{G}^{;p}].\end{array}$$ 
(73)

For the massless conformal case, this reduces to
$$\begin{array}{c}N\left[G\right]=\frac{1}{144}\left\{R{R}^{\prime}{G}^{2}6G\left(R{\square}^{\prime}+{R}^{\prime}\square \right)G+18\left[\left(\square G\right)\left({\square}^{\prime}G\right)+{\square}^{\prime}\square G\right]\right\},\end{array}$$ 
(74)

which holds for any function
$G(x,y)$
. For
$G$
being the Green function, it satisfies the field equation ( 2 ):
$$\begin{array}{c}\square G=({m}^{2}+\xi R)G.\end{array}$$ 
(75)

We will only assume that the Green function satisfies the field equation in its first variable. Using the fact
${\square}^{\prime}R=0$
(because the covariant derivatives act at a different point than at which
$R$
is supported), it follows that
$$\begin{array}{c}{\square}^{\prime}\square G=({m}^{2}+\xi R){\square}^{\prime}G.\end{array}$$ 
(76)

With these results, the noise kernel trace becomes
$$\begin{array}{ccc}N\left[G\right]& =& \frac{1}{2}\left[{m}^{2}\left(13\xi \right)+3R\left(\frac{1}{6}\xi \right)\xi \right]\left[{G}^{2}\left(2{m}^{2}+{R}^{\prime}\xi \right)+\left(16\xi \right){G}_{;{p}^{\prime}}{{G}_{;}}^{{p}^{\prime}}6G\xi {{G}_{;{p}^{\prime}}}^{{p}^{\prime}}\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{2}\left(\frac{1}{6}\xi \right)\left[3\left(2{m}^{2}+{R}^{\prime}\xi \right){G}_{;p}{G}^{;p}18\xi {G}_{;p}{{G}_{;{p}^{\prime}}}^{p{p}^{\prime}}+18\left(\frac{1}{6}\xi \right){G}_{;{p}^{\prime}p}{{G}_{;}}^{{p}^{\prime}p}\right],\end{array}$$ 
(77)

which vanishes for the massless conformal case. We have thus shown, based solely on the definition of the point separated noise kernel, that there is no noise associated with the trace anomaly. This result obtained in [
245]
is completely general since it is assumed that the Green function is only satisfying the field equations in its first variable; an alternative proof of this result was given in [
208]
.
This condition holds not just for the classical field case, but also for the regularized quantum case, where one does not expect the Green function to satisfy the field equation in both variables. One can see this result from the simple observation used in Section
3 : Since the trace anomaly is known to be locally determined and quantum state independent, whereas the noise present in the quantum field is nonlocal, it is hard to find a noise associated with it. This general result is in agreement with previous findings [
44,
167,
58]
, derived from the FeynmanVernon influence functional formalism [
89,
88]
for some particular cases.
6 Metric Fluctuations in Minkowski Spacetime
Although the Minkowski vacuum is an eigenstate of the total fourmomentum operator of a field in Minkowski spacetime, it is not an eigenstate of the stressenergy operator. Hence, even for those solutions of semiclassical gravity such as the Minkowski metric, for which the expectation value of the stressenergy operator can always be chosen to be zero, the fluctuations of this operator are nonvanishing. This fact leads to consider the stochastic metric perturbations induced by these fluctuations.
Here we derive the Einstein–Langevin equation for the metric perturbations in a Minkowski background. We solve this equation for the linearized Einstein tensor and compute the associated twopoint correlation functions, as well as, the twopoint correlation functions for the metric perturbations. Even though, in this case, we expect to have negligibly small values for these correlation functions for points separated by lengths larger than the Planck length, there are several reasons why it is worth carrying out this calculation.
On the one hand, these are the first backreaction solutions of the full Einstein–Langevin equation. There are analogous solutions to a “reduced” version of this equation inspired in a “minisuperspace” model [
59,
38]
, and there is also a previous attempt to obtain a solution to the Einstein–Langevin equation in [
58]
, but there the nonlocal terms in the Einstein–Langevin equation were neglected.
On the other hand, the results of this calculation, which confirm our expectations that gravitational fluctuations are negligible at length scales larger than the Planck length, but also predict that the fluctuations are strongly suppressed on small scales, can be considered a first test of stochastic semiclassical gravity. In addition, these results reveal an important connection between stochastic gravity and the large
$N$
expansion of quantum gravity. We can also extract conclusions on the possible qualitative behavior of the solutions to the Einstein–Langevin equation. Thus, it is interesting to note that the correlation functions at short scales are characterized by correlation lengths of the order of the Planck length; furthermore, such correlation lengths enter in a nonanalytic way in the correlation functions.
We advise the reader that his section is rather technical since it deals with an explicit nontrivial backreaction computation in stochastic gravity. We have tried to make it reasonable selfcontained and detailed, however a more detailed exposition can be found in [
209]
.
6.1 Perturbations around Minkowski spacetime
The Minkowski metric
${\eta}_{ab}$
, in a manifold
$\mathcal{\mathcal{M}}$
which is topologically
$I{R}^{4}$
, and the usual Minkowski vacuum, denoted as
$0\rangle $
, are the class of simplest solutions to the semiclassical Einstein equation ( 7 ), the socalled trivial solutions of semiclassical gravity [
91]
. They constitute the ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in which the renormalized expectation value
$\langle 0\left{\hat{T}}_{R}^{ab}\right[\eta \left]\right0\rangle =0$
. Thus, Minkowski spacetime
$(I{R}^{4},{\eta}_{ab})$
and the vacuum state
$0\rangle $
are a solution to the semiclassical Einstein equation with renormalized cosmological constant
$\Lambda =0$
. The fact that the vacuum expectation value of the renormalized stressenergy operator in Minkowski spacetime should vanish was originally proposed by Wald [
282]
, and it may be understood as a renormalization convention [
100,
113]
. Note that other possible solutions of semiclassical gravity with zero vacuum expectation value of the stressenergy tensor are the exact gravitational plane waves, since they are known to be vacuum solutions of Einstein equations which induce neither particle creation nor vacuum polarization [
107,
73,
104]
.
As we have already mentioned the vacuum
$0t\rangle $
is an eigenstate of the total fourmomentum operator in Minkowski spacetime, but not an eigenstate of
${\hat{T}}_{ab}^{R}\left[\eta \right]$
. Hence, even in the Minkowski background, there are quantum fluctuations in the stressenergy tensor and, as a result, the noise kernel does not vanish. This fact leads to consider the stochastic corrections to this class of trivial solutions of semiclassical gravity. Since, in this case, the Wightman and Feynman functions ( 37 ), their values in the twopoint coincidence limit, and the products of derivatives of two of such functions appearing in expressions ( 38 ) and ( 39 ) are known in dimensional regularization, we can compute the Einstein–Langevin equation using the methods outlined in Sections 3 and 4 .
To perform explicit calculations it is convenient to work in a global inertial coordinate system
$\left\{{x}^{\mu}\right\}$
and in the associated basis, in which the components of the flat metric are simply
${\eta}_{\mu \nu}=diag(1,1,...,1)$
. In Minkowski spacetime, the components of the classical stressenergy tensor ( 3 ) reduce to
$$\begin{array}{c}{T}^{\mu \nu}[\eta ,\phi ]={\partial}^{\mu}\phi {\partial}^{\nu}\phi \frac{1}{2}{\eta}^{\mu \nu}{\partial}^{\rho}\phi {\partial}_{\rho}\phi \frac{1}{2}{\eta}^{\mu \nu}{m}^{2}{\phi}^{2}+\xi \left({\eta}^{\mu \nu}\square {\partial}^{\mu}{\partial}^{\nu}\right){\phi}^{2},\end{array}$$ 
(78)

where
$\square \equiv {\partial}_{\mu}{\partial}^{\mu}$
, and the formal expression for the components of the corresponding “operator” in dimensional regularization, see Equation ( 4 ), is
$$\begin{array}{c}{\hat{T}}_{n}^{\mu \nu}\left[\eta \right]=\frac{1}{2}\left\{{\partial}^{\mu}{\hat{\phi}}_{n},{\partial}^{\nu}{\hat{\phi}}_{n}\right\}+{\mathcal{D}}^{\mu \nu}{\hat{\phi}}_{n}^{2},\end{array}$$ 
(79)

where
${\mathcal{D}}^{\mu \nu}$
is the differential operator ( 5 ), with
${g}_{\mu \nu}={\eta}_{\mu \nu}$
,
${R}_{\mu \nu}=0$
, and
${\nabla}_{\mu}={\partial}_{\mu}$
. The field
${\hat{\phi}}_{n}\left(x\right)$
is the field operator in the Heisenberg representation in an
$n$
dimensional Minkowski spacetime, which satisfies the Klein–Gordon equation ( 2 ). We use here a stressenergy tensor which differs from the canonical one, which corresponds to
$\xi =0$
; both tensors, however, define the same total momentum.
The Wightman and Feynman functions (
37 ) for
${g}_{\mu \nu}={\eta}_{\mu \nu}$
are well known:
$$\begin{array}{c}{G}_{n}^{+}(x,y)=i{\Delta}_{n}^{+}(xy),{G}_{{F}_{n}}(x,y)={\Delta}_{{F}_{n}}(xy),\end{array}$$ 
(80)

with
$$\begin{array}{c}\begin{array}{ccc}{\Delta}_{n}^{+}\left(x\right)& =& 2\pi i\int \frac{{d}^{n}k}{(2\pi {)}^{n}}{e}^{ikx}\delta ({k}^{2}+{m}^{2})\theta \left({k}^{0}\right),\\ {\Delta}_{{F}_{n}}\left(x\right)& =& \int \frac{{d}^{n}k}{(2\pi {)}^{n}}\frac{{e}^{ikx}}{{k}^{2}+{m}^{2}i\epsilon}\text{for}\epsilon \to {0}^{+},\end{array}\end{array}$$ 
(81)

where
${k}^{2}\equiv {\eta}_{\mu \nu}{k}^{\mu}{k}^{\nu}$
and
$kx\equiv {\eta}_{\mu \nu}{k}^{\mu}{x}^{\nu}$
. Note that the derivatives of these functions satisfy
${\partial}_{\mu}^{x}{\Delta}_{n}^{+}(xy)={\partial}_{\mu}{\Delta}_{n}^{+}(xy)$
and
${\partial}_{\mu}^{y}{\Delta}_{n}^{+}(xy)={\partial}_{\mu}{\Delta}_{n}^{+}(xy)$
, and similarly for the Feynman propagator
${\Delta}_{{F}_{n}}(xy)$
. To write down the semiclassical Einstein equation ( 7 ) in
$n$
dimensions for this case, we need to compute the vacuum expectation value of the stressenergy operator components ( 79 ). Since, from ( 80 ), we have that
$\langle 0\left{\hat{\phi}}_{n}^{2}\right(x\left)\right0\rangle =i{\Delta}_{{F}_{n}}\left(0\right)=i{\Delta}_{n}^{+}\left(0\right)$
, which is a constant (independent of
$x$
), we have simply
$$\begin{array}{c}\langle 0\left{\hat{T}}_{n}^{\mu \nu}\left[\eta \right]\right0\rangle =i\int \frac{{d}^{n}k}{(2\pi {)}^{n}}\frac{{k}^{\mu}{k}^{\nu}}{{k}^{2}+{m}^{2}i\epsilon}=\frac{{\eta}^{\mu \nu}}{2}{\left(\frac{{m}^{2}}{4\pi}\right)}^{n/2}\Gamma \left(\frac{n}{2}\right),\end{array}$$ 
(82)

where the integrals in dimensional regularization have been computed in the standard way [
209]
, and where
$\Gamma \left(z\right)$
is Euler's gamma function. The semiclassical Einstein equation ( 7 ) in
$n$
dimensions before renormalization reduces now to
$$\begin{array}{c}\frac{{\Lambda}_{B}}{8\pi {G}_{B}}{\eta}^{\mu \nu}={\mu}^{(n4)}\langle 0\left{\hat{T}}_{n}^{\mu \nu}\left[\eta \right]\right0\rangle .\end{array}$$ 
(83)

This equation, thus, simply sets the value of the bare coupling constant
${\Lambda}_{B}/{G}_{B}$
. Note, from Equation ( 82 ), that in order to have
$\langle 0\left{\hat{T}}_{R}^{\mu \nu}\right0\rangle \left[\eta \right]=0$
, the renormalized and regularized stressenergy tensor “operator” for a scalar field in Minkowski spacetime, see Equation ( 6 ), has to be defined as
$$\begin{array}{c}{\hat{T}}_{R}^{\mu \nu}\left[\eta \right]={\mu}^{(n4)}{\hat{T}}_{n}^{\mu \nu}\left[\eta \right]\frac{{\eta}^{\mu \nu}}{2}\frac{{m}^{4}}{(4\pi {)}^{2}}{\left(\frac{{m}^{2}}{4\pi {\mu}^{2}}\right)}^{\frac{n4}{2}}\Gamma \left(\frac{n}{2}\right),\end{array}$$ 
(84)

which corresponds to a renormalization of the cosmological constant
$$\begin{array}{c}\frac{{\Lambda}_{B}}{{G}_{B}}=\frac{\Lambda}{G}\frac{2}{\pi}\frac{{m}^{4}}{n(n2)}{\kappa}_{n}+\mathcal{O}(n4),\end{array}$$ 
(85)

where
$$\begin{array}{c}{\kappa}_{n}\equiv \frac{1}{n4}{\left(\frac{{e}^{\gamma}{m}^{2}}{4\pi {\mu}^{2}}\right)}^{\frac{n4}{2}}=\frac{1}{n4}+\frac{1}{2}ln\left(\frac{{e}^{\gamma}{m}^{2}}{4\pi {\mu}^{2}}\right)+\mathcal{O}(n4),\end{array}$$ 
(86)

with
$\gamma $
being Euler's constant. In the case of a massless scalar field,
${m}^{2}=0$
, one simply has
${\Lambda}_{B}/{G}_{B}=\Lambda /G$
. Introducing this renormalized coupling constant into Equation ( 83 ), we can take the limit
$n\to 4$
. We find that, for
$(I{R}^{4},{\eta}_{ab},0\rangle )$
to satisfy the semiclassical Einstein equation, we must take
$\Lambda =0$
.
We can now write down the Einstein–Langevin equations for the components
${h}_{\mu \nu}$
of the stochastic metric perturbation in dimensional regularization. In our case, using
$\langle 0\left{\hat{\phi}}_{n}^{2}\right(x\left)\right0\rangle =i{\Delta}_{{F}_{n}}\left(0\right)$
and the explicit expression of Equation ( 34 ), we obtain
$$\begin{array}{ccc}\frac{1}{8\pi {G}_{B}}\left[{G}^{\left(1\right)\mu \nu}+{\Lambda}_{B}\left({h}^{\mu \nu}\frac{1}{2}{\eta}^{\mu \nu}h\right)\right]\left(x\right)\frac{4}{3}{\alpha}_{B}{D}^{\left(1\right)\mu \nu}\left(x\right)2{\beta}_{B}{B}^{\left(1\right)\mu \nu}\left(x\right)& & \end{array}$$  
$$\begin{array}{ccc}\xi {G}^{\left(1\right)\mu \nu}\left(x\right){\mu}^{(n4)}i{\Delta}_{{F}_{n}}\left(0\right)+\frac{1}{2}\int {d}^{n}y{\mu}^{(n4)}{H}_{n}^{\mu \nu \alpha \beta}(x,y){h}_{\alpha \beta}\left(y\right)& =& {\xi}^{\mu \nu}\left(x\right).\end{array}$$ 
(87)

The indices in
${h}_{\mu \nu}$
are raised with the Minkowski metric, and
$h\equiv {h}_{\rho}^{\rho}$
; here a superindex
${}^{\left(1\right)}$
denotes the components of a tensor linearized around the flat metric. Note that in
$n$
dimensions the twopoint correlation functions for the field
${\xi}^{\mu \nu}$
is written as
$$\begin{array}{c}{\langle {\xi}^{\mu \nu}\left(x\right){\xi}^{\alpha \beta}\left(y\right)\rangle}_{s}={\mu}^{2(n4)}{N}_{n}^{\mu \nu \alpha \beta}(x,y).\end{array}$$ 
(88)

Explicit expressions for
${D}^{\left(1\right)\mu \nu}$
and
${B}^{\left(1\right)\mu \nu}$
are given by
$$\begin{array}{c}{D}^{\left(1\right)\mu \nu}\left(x\right)=\frac{1}{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}{h}_{\alpha \beta}\left(x\right),{B}^{\left(1\right)\mu \nu}\left(x\right)=2{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}{h}_{\alpha \beta}\left(x\right),\end{array}$$ 
(89)

with the differential operators
${\mathcal{\mathcal{F}}}_{x}^{\mu \nu}\equiv {\eta}^{\mu \nu}{\square}_{x}{\partial}_{x}^{\mu}{\partial}_{x}^{\nu}$
and
${\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}\equiv 3{\mathcal{\mathcal{F}}}_{x}^{\mu (\alpha}{\mathcal{\mathcal{F}}}_{x}^{\beta )\nu}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}$
.
6.2 The kernels in the Minkowski background
Since the two kernels ( 36 ) are free of ultraviolet divergences in the limit
$n\to 4$
, we can deal directly with the
${F}^{\mu \nu \alpha \beta}(xy)\equiv {lim}_{n\to 4}{\mu}^{2(n4)}{F}_{n}^{\mu \nu \alpha \beta}$
in Equation ( 35 ). The kernels
${N}^{\mu \nu \alpha \beta}(x,y)=Re{F}^{\mu \nu \alpha \beta}(xy)$
and
${H}_{A}^{\mu \nu \alpha \beta}(x,y)=Im{F}^{\mu \nu \alpha \beta}(xy)$
are actually the components of the “physical” noise and dissipation kernels that will appear in the Einstein–Langevin equations once the renormalization procedure has been carried out. The bitensor
${F}^{\mu \nu \alpha \beta}$
can be expressed in terms of the Wightman function in four spacetime dimensions, according to Equation ( 38 ). The different terms in this kernel can be easily computed using the integrals
$$\begin{array}{c}I\left(p\right)\equiv \int \frac{{d}^{4}k}{(2\pi {)}^{4}}\delta ({k}^{2}+{m}^{2})\theta ({k}^{0})\delta \left[\right(kp{)}^{2}+{m}^{2}\left]\theta \right({k}^{0}{p}^{0}),\end{array}$$ 
(90)

and
${I}^{{\mu}_{1}...{\mu}_{r}}\left(p\right)$
, which are defined as the previous one by inserting the momenta
${k}^{{\mu}_{1}}...{k}^{{\mu}_{r}}$
with
$r=1,...,4$
in the integrand. All these integrals can be expressed in terms of
$I\left(p\right)$
; see [
209]
for the explicit expressions. It is convenient to separate
$I\left(p\right)$
into its even and odd parts with respect to the variables
${p}^{\mu}$
as
$$\begin{array}{c}I\left(p\right)={I}_{S}\left(p\right)+{I}_{A}\left(p\right),\end{array}$$ 
(91)

where
${I}_{S}(p)={I}_{S}\left(p\right)$
and
${I}_{A}(p)={I}_{A}\left(p\right)$
. These two functions are explicitly given by
$$\begin{array}{c}\begin{array}{ccc}{I}_{S}\left(p\right)& =& \frac{1}{8(2\pi {)}^{3}}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}},\\ {I}_{A}\left(p\right)& =& \frac{1}{8(2\pi {)}^{3}}sign{p}^{0}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}.\end{array}\end{array}$$ 
(92)

After some manipulations, we find
$$\begin{array}{ccc}{F}^{\mu \nu \alpha \beta}\left(x\right)& =& \frac{{\pi}^{2}}{45}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}I\left(p\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{8{\pi}^{2}}{9}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}I\left(p\right),\end{array}$$ 
(93)

where
$\Delta \xi \equiv \xi \frac{1}{6}$
. The real and imaginary parts of the last expression, which yield the noise and dissipation kernels, are easily recognized as the terms containing
${I}_{S}\left(p\right)$
and
${I}_{A}\left(p\right)$
, respectively. To write them explicitly, it is useful to introduce the new kernels
$$\begin{array}{c}\begin{array}{ccc}{N}_{A}(x;{m}^{2})& \equiv & \frac{1}{480\pi}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2},\\ {N}_{B}(x;{m}^{2},\Delta \xi )& \equiv & \frac{1}{72\pi}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2},\\ {D}_{A}(x;{m}^{2})& \equiv & \frac{i}{480\pi}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}sign{p}^{0}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2},\\ {D}_{B}(x;{m}^{2},\Delta \xi )& \equiv & \frac{i}{72\pi}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}sign{p}^{0}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2},\end{array}\end{array}$$ 
(94)

and we finally get
$$\begin{array}{c}\begin{array}{ccc}{N}^{\mu \nu \alpha \beta}(x,y)& =& \frac{1}{6}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}{N}_{A}(xy;{m}^{2})+{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}{N}_{B}(xy;{m}^{2},\Delta \xi ),\\ {H}_{A}^{\mu \nu \alpha \beta}(x,y)& =& \frac{1}{6}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}{D}_{A}(xy;{m}^{2})+{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}{D}_{B}(xy;{m}^{2},\Delta \xi ).\end{array}\end{array}$$ 
(95)

Notice that the noise and dissipation kernels defined in Equation ( 94 ) are actually real because, for the noise kernels, only the
$cospx$
terms of the exponentials
${e}^{ipx}$
contribute to the integrals, and, for the dissipation kernels, the only contribution of such exponentials comes from the
$isinpx$
terms.
The evaluation of the kernel
${H}_{{S}_{n}}^{\mu \nu \alpha \beta}(x,y)$
is a more involved task. Since this kernel contains divergences in the limit
$n\to 4$
, we use dimensional regularization. Using Equation ( 39 ), this kernel can be written in terms of the Feynman propagator ( 81 ) as
$$\begin{array}{c}{\mu}^{(n4)}{H}_{{S}_{n}}^{\mu \nu \alpha \beta}(x,y)=Im{K}^{\mu \nu \alpha \beta}(xy),\end{array}$$ 
(96)

where
$$\begin{array}{ccc}{K}^{\mu \nu \alpha \beta}\left(x\right)\equiv {\mu}^{(n4)}& \{& 2{\partial}^{\mu}{\partial}^{(\alpha}{\Delta}_{{F}_{n}}\left(x\right){\partial}^{\beta )}{\partial}^{\nu}{\Delta}_{{F}_{n}}\left(x\right)+2{\mathcal{D}}^{\mu \nu}\left({\partial}^{\alpha}{\Delta}_{{F}_{n}}\left(x\right){\partial}^{\beta}{\Delta}_{{F}_{n}}\left(x\right)\right)\end{array}$$  
$$\begin{array}{ccc}& & +2{\mathcal{D}}^{\alpha \beta}\left({\partial}^{\mu}{\Delta}_{{F}_{n}}\left(x\right){\partial}^{\nu}{\Delta}_{{F}_{n}}\left(x\right)\right)+2{\mathcal{D}}^{\mu \nu}{\mathcal{D}}^{\alpha \beta}\left({\Delta}_{{F}_{n}}^{2}\left(x\right)\right)\end{array}$$  
$$\begin{array}{ccc}& & +[{\eta}^{\mu \nu}{\partial}^{(\alpha}{\Delta}_{{F}_{n}}\left(x\right){\partial}^{\beta )}+{\eta}^{\alpha \beta}{\partial}^{(\mu}{\Delta}_{{F}_{n}}\left(x\right){\partial}^{\nu )}+{\Delta}_{{F}_{n}}\left(0\right)\left({\eta}^{\mu \nu}{\mathcal{D}}^{\alpha \beta}+{\eta}^{\alpha \beta}{\mathcal{D}}^{\mu \nu}\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{4}{\eta}^{\mu \nu}{\eta}^{\alpha \beta}\left({\Delta}_{{F}_{n}}\left(x\right)\square {m}^{2}{\Delta}_{{F}_{n}}\left(0\right)\right)]{\delta}^{n}\left(x\right)\}.\end{array}$$ 
(97)

Let us define the integrals
$$\begin{array}{c}{J}_{n}\left(p\right)\equiv {\mu}^{(n4)}\int \frac{{d}^{n}k}{(2\pi {)}^{n}}\frac{1}{({k}^{2}+{m}^{2}i\epsilon )\left[\right(kp{)}^{2}+{m}^{2}i\epsilon ]},\end{array}$$ 
(98)

and
${J}_{n}^{{\mu}_{1}...{\mu}_{r}}\left(p\right)$
obtained by inserting the momenta
${k}^{{\mu}_{1}}...{k}^{{\mu}_{r}}$
into the previous integral, together with
$$\begin{array}{c}{I}_{{0}_{n}}\equiv {\mu}^{(n4)}\int \frac{{d}^{n}k}{(2\pi {)}^{n}}\frac{1}{({k}^{2}+{m}^{2}i\epsilon )},\end{array}$$ 
(99)

and
${I}_{{0}_{n}}^{{\mu}_{1}...{\mu}_{r}}$
which are also obtained by inserting momenta in the integrand. Then, the different terms in Equation ( 97 ) can be computed; these integrals are explicitly given in [
209]
. It is found that
${I}_{{0}_{n}}^{\mu}=0$
, and the remaining integrals can be written in terms of
${I}_{{0}_{n}}$
and
${J}_{n}\left(p\right)$
. It is useful to introduce the projector
${P}^{\mu \nu}$
orthogonal to
${p}^{\mu}$
and the tensor
${P}^{\mu \nu \alpha \beta}$
as
$$\begin{array}{c}{p}^{2}{P}^{\mu \nu}\equiv {\eta}^{\mu \nu}{p}^{2}{p}^{\mu}{p}^{\nu},{P}^{\mu \nu \alpha \beta}\equiv 3{P}^{\mu (\alpha}{P}^{\beta )\nu}{P}^{\mu \nu}{P}^{\alpha \beta}.\end{array}$$ 
(100)

Then the action of the operator
${\mathcal{\mathcal{F}}}_{x}^{\mu \nu}$
is simply written as
${\mathcal{\mathcal{F}}}_{x}^{\mu \nu}\int {d}^{n}p{e}^{ipx}f\left(p\right)=\int {d}^{n}p{e}^{ipx}f\left(p\right){p}^{2}{P}^{\mu \nu}$
, where
$f\left(p\right)$
is an arbitrary function of
${p}^{\mu}$
.
Finally after a rather long but straightforward calculation, and after expanding around
$n=4$
, we get,
$$\begin{array}{ccc}{K}^{\mu \nu \alpha \beta}\left(x\right)=\frac{i}{(4\pi {)}^{2}}& \{& {\kappa}_{n}[\frac{1}{90}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}{\delta}^{n}\left(x\right)+4\Delta {\xi}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}{\delta}^{n}\left(x\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{2}{3}\frac{{m}^{2}}{(n2)}({\eta}^{\mu \nu}{\eta}^{\alpha \beta}{\square}_{x}{\eta}^{\mu (\alpha}{\eta}^{\beta )\nu}{\square}_{x}+{\eta}^{\mu (\alpha}{\partial}_{x}^{\beta )}{\partial}_{x}^{\nu}\end{array}$$  
$$\begin{array}{ccc}& & +{\eta}^{\nu (\alpha}{\partial}_{x}^{\beta )}{\partial}_{x}^{\mu}{\eta}^{\mu \nu}{\partial}_{x}^{\alpha}{\partial}_{x}^{\beta}{\eta}^{\alpha \beta}{\partial}_{x}^{\mu}{\partial}_{x}^{\nu}){\delta}^{n}\left(x\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{4{m}^{4}}{n(n2)}(2{\eta}^{\mu (\alpha}{\eta}^{\beta )\nu}{\eta}^{\mu \nu}{\eta}^{\alpha \beta}){\delta}^{n}\left(x\right)]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{180}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}\int \frac{{d}^{n}p}{(2\pi {)}^{n}}{e}^{ipx}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}\overline{\phi}\left({p}^{2}\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{2}{9}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}\int \frac{{d}^{n}p}{(2\pi {)}^{n}}{e}^{ipx}{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}\overline{\phi}\left({p}^{2}\right)\end{array}$$  
$$\begin{array}{ccc}& & \left[\frac{4}{675}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}+\frac{1}{270}(60\xi 11){\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}\right]{\delta}^{n}\left(x\right)\end{array}$$  
$$\begin{array}{ccc}& & {m}^{2}\left[\frac{2}{135}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}+\frac{1}{27}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}\right]{\Delta}_{n}\left(x\right)\}+\mathcal{O}(n4),\end{array}$$ 
(101)

where
${\kappa}_{n}$
has been defined in Equation ( 86 ), and
$\overline{\phi}\left({p}^{2}\right)$
and
${\Delta}_{n}\left(x\right)$
are given by
$$\begin{array}{ccc}\overline{\phi}\left({p}^{2}\right)& \equiv & {\int}_{0}^{1}d\alpha ln\left(1+\frac{{p}^{2}}{{m}^{2}}\alpha (1\alpha )i\epsilon \right)=i\pi \theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}+\phi \left({p}^{2}\right),\end{array}$$ 
(102)

$$\begin{array}{ccc}{\Delta}_{n}\left(x\right)& \equiv & \int \frac{{d}^{n}p}{(2\pi {)}^{n}}{e}^{ipx}\frac{1}{{p}^{2}},\end{array}$$ 
(103)

where
$$\phi \left({p}^{2}\right)\equiv {\int}_{0}^{1}d\alpha ln\left1+\frac{{p}^{2}}{{m}^{2}}\alpha (1\alpha )\right.$$
The imaginary part of Equation ( 101 ) gives the kernel components
${\mu}^{(n4)}{H}_{{S}_{n}}^{\mu \nu \alpha \beta}(x,y)$
, according to Equation ( 96 ). It can be easily obtained multiplying this expression by
$i$
and retaining only the real part
$\phi \left({p}^{2}\right)$
of the function
$\overline{\phi}\left({p}^{2}\right)$
.
6.3 The Einstein–Langevin equation
With the previous results for the kernels we can now write the
$n$
dimensional Einstein–Langevin equation ( 87 ), previous to the renormalization. Taking also into account Equations ( 82 ) and ( 83 ), we may finally write:
$$\begin{array}{ccc}& & \frac{1}{8\pi {G}_{B}}{G}^{\left(1\right)\mu \nu}\left(x\right)\frac{4}{3}{\alpha}_{B}{D}^{\left(1\right)\mu \nu}\left(x\right)2{\beta}_{B}{B}^{\left(1\right)\mu \nu}\left(x\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{{\kappa}_{n}}{(4\pi {)}^{2}}\left[4\Delta \xi \frac{{m}^{2}}{(n2)}{G}^{\left(1\right)\mu \nu}+\frac{1}{90}{D}^{\left(1\right)\mu \nu}\Delta {\xi}^{2}{B}^{\left(1\right)\mu \nu}\right]\left(x\right)\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{2880{\pi}^{2}}\{\frac{16}{15}{D}^{\left(1\right)\mu \nu}\left(x\right)+\left(\frac{1}{6}10\Delta \xi \right){B}^{\left(1\right)\mu \nu}\left(x\right)\end{array}$$  
$$\begin{array}{ccc}& & +\int {d}^{n}y\int \frac{{d}^{n}p}{(2\pi {)}^{n}}{e}^{ip(xy)}\phi \left({p}^{2}\right)\left[{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}{D}^{\left(1\right)\mu \nu}\left(y\right)+10{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}{B}^{\left(1\right)\mu \nu}\left(y\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & \frac{{m}^{2}}{3}\int {d}^{n}y{\Delta}_{n}(xy)\left(8{D}^{\left(1\right)\mu \nu}+5{B}^{\left(1\right)\mu \nu}\right)\left(y\right)\}\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{2}\int {d}^{n}y{\mu}^{(n4)}{H}_{{A}_{n}}^{\mu \nu \alpha \beta}(x,y){h}_{\alpha \beta}\left(y\right)+\mathcal{O}(n4)={\xi}^{\mu \nu}\left(x\right).\end{array}$$ 
(104)

Notice that the terms containing the bare cosmological constant have cancelled. These equations can now be renormalized, that is, we can now write the bare coupling constants as renormalized coupling constants plus some suitably chosen counterterms, and take the limit
$n\to 4$
. In order to carry out such a procedure, it is convenient to distinguish between massive and massless scalar fields. The details of the calculation can be found in [
209]
.
It is convenient to introduce the two new kernels
$$\begin{array}{c}\begin{array}{ccc}{H}_{A}(x;{m}^{2})& \equiv & \frac{1}{480{\pi}^{2}}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}\\ & & \times \left\{{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}\left[i\pi sign{p}^{0}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}+\phi \left({p}^{2}\right)\right]\frac{8}{3}\frac{{m}^{2}}{{p}^{2}}\right\},\\ {H}_{B}(x;{m}^{2},\Delta \xi )& \equiv & \frac{1}{72{\pi}^{2}}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}\\ & & \times \left\{{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}\left[i\pi sign{p}^{0}\theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}+\phi \left({p}^{2}\right)\right]\frac{1}{6}\frac{{m}^{2}}{{p}^{2}}\right\},\end{array}\end{array}$$ 
(105)

where
$\phi \left({p}^{2}\right)$
is given by the restriction to
$n=4$
of expression ( 102 ). The renormalized coupling constants
$1/G$
,
$\alpha $
, and
$\beta $
are easily computed as it was done in Equation ( 85 ). Substituting their expressions into Equation ( 104 ), we can take the limit
$n\to 4$
, using the fact that, for
$n=4$
,
${D}^{\left(1\right)\mu \nu}\left(x\right)=\frac{3}{2}{A}^{\left(1\right)\mu \nu}\left(x\right)$
, we obtain the corresponding semiclassical Einstein–Langevin equation.
For the massless case one needs the limit
$m\to 0$
of Equation ( 104 ). In this case it is convenient to separate
${\kappa}_{n}$
in Equation ( 86 ) as
${\kappa}_{n}={\stackrel{~}{\kappa}}_{n}+\frac{1}{2}ln({m}^{2}/{\mu}^{2})+\mathcal{O}(n4)$
, where
$$\begin{array}{c}{\stackrel{~}{\kappa}}_{n}\equiv \frac{1}{n4}{\left(\frac{{e}^{\gamma}}{4\pi}\right)}^{\frac{n4}{2}}=\frac{1}{n4}+\frac{1}{2}ln\left(\frac{{e}^{\gamma}}{4\pi}\right)+\mathcal{O}(n4),\end{array}$$ 
(106)

and use that, from Equation ( 102 ), we have
$$\begin{array}{c}{lim}_{{m}^{2}\to 0}\left[\phi \left({p}^{2}\right)+ln\frac{{m}^{2}}{{\mu}^{2}}\right]=2+ln\left\frac{{p}^{2}}{{\mu}^{2}}\right.\end{array}$$ 
(107)

The coupling constants are then easily renormalized. We note that in the massless limit, the Newtonian gravitational constant is not renormalized and, in the conformal coupling case,
$\Delta \xi =0$
, we have that
${\beta}_{B}=\beta $
. Note also that, by making
$m=0$
in Equation ( 94 ), the noise and dissipation kernels can be written as
$$\begin{array}{c}\begin{array}{cccccc}{N}_{A}(x;{m}^{2}=0)& =& N\left(x\right),& {N}_{B}(x;{m}^{2}=0,\Delta \xi )& =& 60\Delta {\xi}^{2}N\left(x\right),\\ {D}_{A}(x;{m}^{2}=0)& =& D\left(x\right),& {D}_{B}(x;{m}^{2}=0,\Delta \xi )& =& 60\Delta {\xi}^{2}D\left(x\right),\end{array}\end{array}$$ 
(108)

where
$$\begin{array}{c}N\left(x\right)\equiv \frac{1}{480\pi}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}\theta ({p}^{2}),D\left(x\right)\equiv \frac{i}{480\pi}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}sign{p}^{0}\theta ({p}^{2}).\end{array}$$ 
(109)

It is also convenient to introduce the new kernel
$$\begin{array}{ccc}H(x;{\mu}^{2})& \equiv & \frac{1}{480{\pi}^{2}}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}\left[ln\left\frac{{p}^{2}}{{\mu}^{2}}\righti\pi sign{p}^{0}\theta ({p}^{2})\right]\end{array}$$  
$$\begin{array}{ccc}& =& \frac{1}{480{\pi}^{2}}{lim}_{\epsilon \to {0}^{+}}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}{e}^{ipx}ln\left(\frac{({p}^{0}+i\epsilon {)}^{2}+{p}^{i}{p}_{i}}{{\mu}^{2}}\right).\end{array}$$ 
(110)

This kernel is real and can be written as the sum of an even part and an odd part in the variables
${x}^{\mu}$
, where the odd part is the dissipation kernel
$D\left(x\right)$
. The Fourier transforms ( 109 ) and ( 110 ) can actually be computed and, thus, in this case we have explicit expressions for the kernels in position space; see, for instance, [
179,
56,
137]
.
Finally, the Einstein–Langevin equation for the physical stochastic perturbations
${h}_{\mu \nu}$
can be written in both cases, for
$m\ne 0$
and for
$m=0$
, as
$$\begin{array}{ccc}\frac{1}{8\pi G}{G}^{\left(1\right)\mu \nu}\left(x\right)2\left(\overline{\alpha}{A}^{\left(1\right)\mu \nu}\left(x\right)+\overline{\beta}{B}^{\left(1\right)\mu \nu}\left(x\right)\right)& & \end{array}$$  
$$\begin{array}{ccc}+\frac{1}{4}\int {d}^{4}y\left[{H}_{A}(xy){A}^{\left(1\right)\mu \nu}\left(y\right)+{H}_{B}(xy){B}^{\left(1\right)\mu \nu}\left(y\right)\right]& =& {\xi}^{\mu \nu}\left(x\right),\end{array}$$ 
(111)

where in terms of the renormalized constants
$\alpha $
and
$\beta $
the new constants are
$\overline{\alpha}=\alpha +(3600{\pi}^{2}{)}^{1}$
and
$\overline{\beta}=\beta (\frac{1}{12}5\Delta \xi )(2880{\pi}^{2}{)}^{1}$
. The kernels
${H}_{A}\left(x\right)$
and
${H}_{B}\left(x\right)$
are given by Equations ( 105 ) when
$m\ne 0$
, and
${H}_{A}\left(x\right)=H(x;{\mu}^{2})$
,
${H}_{B}\left(x\right)=60\Delta {\xi}^{2}H(x;{\mu}^{2})$
when
$m=0$
. In the massless case, we can use the arbitrariness of the mass scale
$\mu $
to eliminate one of the parameters
$\overline{\alpha}$
or
$\overline{\beta}$
.
The components of the Gaussian stochastic source
${\xi}^{\mu \nu}$
have zero mean value, and their twopoint correlation functions are given by
$\langle {\xi}^{\mu \nu}\left(x\right){\xi}^{\alpha \beta}\left(y\right){\rangle}_{s}={N}^{\mu \nu \alpha \beta}(x,y)$
, where the noise kernel is given in Equation ( 95 ), which in the massless case reduces to Equation ( 108 ).
It is interesting to consider the massless conformally coupled scalar field, i.e., the case
$\Delta \xi =0$
, which is of particular interest because of its similarities with the electromagnetic field, and also because of its interest in cosmology: Massive fields become conformally invariant when their masses are negligible compared to the spacetime curvature. We have already mentioned that for a conformally coupled field, the stochastic source tensor must be traceless (up to first order in perturbation theory around semiclassical gravity), in the sense that the stochastic variable
${\xi}_{\mu}^{\mu}\equiv {\eta}_{\mu \nu}{\xi}^{\mu \nu}$
behaves deterministically as a vanishing scalar field. This can be directly checked by noticing, from Equations ( 95 ) and ( 108 ), that, when
$\Delta \xi =0$
, one has
$\langle {\xi}_{\mu}^{\mu}\left(x\right){\xi}^{\alpha \beta}\left(y\right){\rangle}_{s}=0$
, since
${\mathcal{\mathcal{F}}}_{\mu}^{\mu}=3\square $
and
${\mathcal{\mathcal{F}}}^{\mu \alpha}{\mathcal{\mathcal{F}}}_{\mu}^{\beta}=\square {\mathcal{\mathcal{F}}}^{\alpha \beta}$
. The Einstein–Langevin equations for this particular case (and generalized to a spatially flat Robertson–Walker background) were first obtained in [
58]
, where the coupling constant
$\beta $
was fixed to be zero. See also [
169]
for a discussion of this result and its connection to the problem of structure formation in the trace anomaly driven inflation [
269,
280,
132]
.
Note that the expectation value of the renormalized stressenergy tensor for a scalar field can be obtained by comparing Equation (
111 ) with the Einstein–Langevin equation ( 14 ), its explicit expression is given in [
209]
. The results agree with the general form found by Horowitz [
137,
138]
using an axiomatic approach, and coincides with that given in [
91]
. The particular cases of conformal coupling,
$\Delta \xi =0$
, and minimal coupling,
$\Delta \xi =1/6$
, are also in agreement with the results for these cases given in [
137,
138,
270,
57,
182]
, modulo local terms proportional to
${A}^{\left(1\right)\mu \nu}$
and
${B}^{\left(1\right)\mu \nu}$
due to different choices of the renormalization scheme. For the case of a massive minimally coupled scalar field,
$\Delta \xi =\frac{1}{6}$
, our result is equivalent to that of [
276]
.
6.4 Correlation functions for gravitational perturbations
Here we solve the Einstein–Langevin equations ( 111 ) for the components
${G}^{\left(1\right)\mu \nu}$
of the linearized Einstein tensor. Then we use these solutions to compute the corresponding twopoint correlation functions, which give a measure of the gravitational fluctuations predicted by the stochastic semiclassical theory of gravity in the present case. Since the linearized Einstein tensor is invariant under gauge transformations of the metric perturbations, these twopoint correlation functions are also gauge invariant. Once we have computed the twopoint correlation functions for the linearized Einstein tensor, we find the solutions for the metric perturbations and compute the associated twopoint correlation functions. The procedure used to solve the Einstein–Langevin equation is similar to the one used by Horowitz [
137]
(see also [
91]
) to analyze the stability of Minkowski spacetime in semiclassical gravity.
We first note that the tensors
${A}^{\left(1\right)\mu \nu}$
and
${B}^{\left(1\right)\mu \nu}$
can be written in terms of
${G}^{\left(1\right)\mu \nu}$
as
$$\begin{array}{c}{A}^{\left(1\right)\mu \nu}=\frac{2}{3}\left({\mathcal{\mathcal{F}}}^{\mu \nu}{{G}^{\left(1\right)}}_{\alpha}^{\alpha}{\mathcal{\mathcal{F}}}_{\alpha}^{\alpha}{G}^{\left(1\right)\mu \nu}\right),{B}^{\left(1\right)\mu \nu}=2{\mathcal{\mathcal{F}}}^{\mu \nu}{{G}^{\left(1\right)}}_{\alpha}^{\alpha},\end{array}$$ 
(112)

where we have used that
$3\square ={\mathcal{\mathcal{F}}}_{\alpha}^{\alpha}$
. Therefore, the Einstein–Langevin equation ( 111 ) can be seen as a linear integrodifferential stochastic equation for the components
${G}^{\left(1\right)\mu \nu}$
. In order to find solutions to Equation ( 111 ), it is convenient to Fourier transform. With the convention
$\stackrel{~}{f}\left(p\right)=\int {d}^{4}x{e}^{ipx}f\left(x\right)$
for a given field
$f\left(x\right)$
, one finds, from Equation ( 112 ),
$$\begin{array}{ccc}{\stackrel{~}{A}}^{\left(1\right)\mu \nu}\left(p\right)& =& 2{p}^{2}{\stackrel{~}{G}}^{\left(1\right)\mu \nu}\left(p\right)\frac{2}{3}{p}^{2}{P}^{\mu \nu}{{\stackrel{~}{G}}^{\left(1\right)}}_{\alpha}^{\alpha}\left(p\right),\end{array}$$  
$$\begin{array}{ccc}{\stackrel{~}{B}}^{\left(1\right)\mu \nu}\left(p\right)& =& 2{p}^{2}{P}^{\mu \nu}{{\stackrel{~}{G}}^{\left(1\right)}}_{\alpha}^{\alpha}\left(p\right).\end{array}$$ 
(113)

The Fourier transform of the Einstein–Langevin Equation ( 111 ) now reads
$$\begin{array}{c}{F}_{\alpha \beta}^{\mu \nu}\left(p\right){\stackrel{~}{G}}^{\left(1\right)\alpha \beta}\left(p\right)=8\pi G{\stackrel{~}{\xi}}^{\mu \nu}\left(p\right),\end{array}$$ 
(114)

where
$$\begin{array}{c}{F}_{\alpha \beta}^{\mu \nu}\left(p\right)\equiv {F}_{1}\left(p\right){\delta}_{(\alpha}^{\mu}{\delta}_{\beta )}^{\nu}+{F}_{2}\left(p\right){p}^{2}{P}^{\mu \nu}{\eta}_{\alpha \beta},\end{array}$$ 
(115)

with
$$\begin{array}{c}\begin{array}{ccc}{F}_{1}\left(p\right)& \equiv & 1+16\pi G{p}^{2}\left[\frac{1}{4}{\stackrel{~}{H}}_{A}\left(p\right)2\overline{\alpha}\right],\\ {F}_{2}\left(p\right)& \equiv & \frac{16}{3}\pi G\left[\frac{1}{4}{\stackrel{~}{H}}_{A}\left(p\right)+\frac{3}{4}{\stackrel{~}{H}}_{B}\left(p\right)2\overline{\alpha}6\overline{\beta}\right].\end{array}\end{array}$$ 
(116)

In the Fourier transformed Einstein–Langevin Equation ( 114 ),
${\stackrel{~}{\xi}}^{\mu \nu}\left(p\right)$
, the Fourier transform of
${\xi}^{\mu \nu}\left(x\right)$
, is a Gaussian stochastic source of zero average, and
$$\begin{array}{c}{\langle {\stackrel{~}{\xi}}^{\mu \nu}\left(p\right){\stackrel{~}{\xi}}^{\alpha \beta}\left({p}^{\prime}\right)\rangle}_{s}=\left(2\pi {)}^{4}{\delta}^{4}\right(p+{p}^{\prime}\left){\stackrel{~}{N}}^{\mu \nu \alpha \beta}\right(p),\end{array}$$ 
(117)

where we have introduced the Fourier transform of the noise kernel. The explicit expression for
${\stackrel{~}{N}}^{\mu \nu \alpha \beta}\left(p\right)$
is found from Equations ( 95 ) and ( 94 ) to be
$$\begin{array}{ccc}{\stackrel{~}{N}}^{\mu \nu \alpha \beta}\left(p\right)=\frac{\theta ({p}^{2}4{m}^{2})}{720\pi}\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}\left[\frac{1}{4}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}({p}^{2}{)}^{2}{P}^{\mu \nu \alpha \beta}+10{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}({p}^{2}{)}^{2}{P}^{\mu \nu}{P}^{\alpha \beta}\right],& & \end{array}$$  
$$\begin{array}{ccc}& & \end{array}$$ 
(118)

which in the massless case reduces to
$$\begin{array}{c}{lim}_{m\to 0}{\stackrel{~}{N}}^{\mu \nu \alpha \beta}\left(p\right)=\frac{1}{480\pi}\theta ({p}^{2})\left[\frac{1}{6}({p}^{2}{)}^{2}{P}^{\mu \nu \alpha \beta}+60\Delta {\xi}^{2}({p}^{2}{)}^{2}{P}^{\mu \nu}{P}^{\alpha \beta}\right].\end{array}$$ 
(119)

6.4.1 Correlation functions for the linearized Einstein tensor
In general, we can write
${G}^{\left(1\right)\mu \nu}=\langle {G}^{\left(1\right)\mu \nu}{\rangle}_{s}+{G}_{f}^{\left(1\right)\mu \nu}$
, where
${G}_{f}^{\left(1\right)\mu \nu}$
is a solution to Equations ( 111 ) with zero average, or Equation ( 114 ) in the Fourier transformed version. The averages
$\langle {G}^{\left(1\right)\mu \nu}{\rangle}_{s}$
must be a solution of the linearized semiclassical Einstein equations obtained by averaging Equations ( 111 ) or ( 114 ). Solutions to these equations (specially in the massless case,
$m=0$
) have been studied by several authors [
137,
138,
141,
247,
248,
273,
274,
128,
262,
182,
91]
, particularly in connection with the problem of the stability of the ground state of semiclassical gravity. The twopoint correlation functions for the linearized Einstein tensor are defined by
$$\begin{array}{ccc}{\mathcal{G}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})& \equiv & {\langle {G}^{\left(1\right)\mu \nu}\left(x\right){G}^{\left(1\right)\alpha \beta}\left({x}^{\prime}\right)\rangle}_{s}{\langle {G}^{\left(1\right)\mu \nu}\left(x\right)\rangle}_{s}{\langle {G}^{\left(1\right)\alpha \beta}\left({x}^{\prime}\right)\rangle}_{s}\end{array}$$  
$$\begin{array}{ccc}& =& {\langle {G}_{f}^{\left(1\right)\mu \nu}\left(x\right){G}_{f}^{\left(1\right)\alpha \beta}\left({x}^{\prime}\right)\rangle}_{s}.\end{array}$$ 
(120)

Now we shall seek the family of solutions to the Einstein–Langevin equation which can be written as a linear functional of the stochastic source, and whose Fourier transform
${\stackrel{~}{G}}^{\left(1\right)\mu \nu}\left(p\right)$
depends locally on
${\stackrel{~}{\xi}}^{\alpha \beta}\left(p\right)$
. Each of such solutions is a Gaussian stochastic field and, thus, it can be completely characterized by the averages
$\langle {G}^{\left(1\right)\mu \nu}{\rangle}_{s}$
and the twopoint correlation functions ( 120 ).
For such a family of solutions,
${\stackrel{~}{G}}_{f}^{\left(1\right)\mu \nu}\left(p\right)$
is the most general solution to Equation ( 114 ) which is linear, homogeneous, and local in
${\stackrel{~}{\xi}}^{\alpha \beta}\left(p\right)$
. It can be written as
$$\begin{array}{c}{\stackrel{~}{G}}_{f}^{\left(1\right)\mu \nu}\left(p\right)=8\pi G{D}_{\alpha \beta}^{\mu \nu}\left(p\right){\stackrel{~}{\xi}}^{\alpha \beta}\left(p\right),\end{array}$$ 
(121)

where
${D}_{\alpha \beta}^{\mu \nu}\left(p\right)$
are the components of a Lorentz invariant tensor field distribution in Minkowski spacetime^{
$\text{3}$
}
, symmetric under the interchanges
$\alpha \leftrightarrow \beta $
and
$\mu \leftrightarrow \nu $
, which is the most general solution of
$$\begin{array}{c}{F}_{\rho \sigma}^{\mu \nu}\left(p\right){D}_{\alpha \beta}^{\rho \sigma}\left(p\right)={\delta}_{(\alpha}^{\mu}{\delta}_{\beta )}^{\nu}.\end{array}$$ 
(122)

In addition, we must impose the conservation condition,
${p}_{\nu}{\stackrel{~}{G}}_{f}^{\left(1\right)\mu \nu}\left(p\right)=0$
, where this zero must be understood as a stochastic variable which behaves deterministically as a zero vector field. We can write
${D}_{\alpha \beta}^{\mu \nu}\left(p\right)={{D}_{p}^{\mu \nu}}_{\alpha \beta}\left(p\right)+{{D}_{h}^{\mu \nu}}_{\alpha \beta}\left(p\right)$
, where
${{D}_{p}^{\mu \nu}}_{\alpha \beta}\left(p\right)$
is a particular solution to Equation ( 122 ) and
${{D}_{h}^{\mu \nu}}_{\alpha \beta}\left(p\right)$
is the most general solution to the homogeneous equation. Consequently, see Equation ( 121 ), we can write
${\stackrel{~}{G}}_{f}^{\left(1\right)\mu \nu}\left(p\right)={\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right)+{\stackrel{~}{G}}_{h}^{\left(1\right)\mu \nu}\left(p\right)$
. To find the particular solution, we try an ansatz of the form
$$\begin{array}{c}{{D}_{p}^{\mu \nu}}_{\alpha \beta}\left(p\right)={d}_{1}\left(p\right){\delta}_{(\alpha}^{\mu}{\delta}_{\beta )}^{\nu}+{d}_{2}\left(p\right){p}^{2}{P}^{\mu \nu}{\eta}_{\alpha \beta}.\end{array}$$ 
(123)

Substituting this ansatz into Equations ( 122 ), it is easy to see that it solves these equations if we take
$$\begin{array}{c}{d}_{1}\left(p\right)={\left[\frac{1}{{F}_{1}\left(p\right)}\right]}_{r},{d}_{2}\left(p\right)={\left[\frac{{F}_{2}\left(p\right)}{{F}_{1}\left(p\right){F}_{3}\left(p\right)}\right]}_{r},\end{array}$$ 
(124)

with
$$\begin{array}{c}{F}_{3}\left(p\right)\equiv {F}_{1}\left(p\right)+3{p}^{2}{F}_{2}\left(p\right)=148\pi G{p}^{2}\left[\frac{1}{4}{\stackrel{~}{H}}_{B}\left(p\right)2\overline{\beta}\right],\end{array}$$ 
(125)

and where the notation
$[{]}_{r}$
means that the zeros of the denominators are regulated with appropriate prescriptions in such a way that
${d}_{1}\left(p\right)$
and
${d}_{2}\left(p\right)$
are well defined Lorentz invariant scalar distributions. This yields a particular solution to the Einstein–Langevin equations,
$$\begin{array}{c}{\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right)=8\pi G{{D}_{p}^{\mu \nu}}_{\alpha \beta}\left(p\right){\stackrel{~}{\xi}}^{\alpha \beta}\left(p\right),\end{array}$$ 
(126)

which, since the stochastic source is conserved, satisfies the conservation condition. Note that, in the case of a massless scalar field (
$m=0$
), the above solution has a functional form analogous to that of the solutions of linearized semiclassical gravity found in the appendix of [
91]
. Notice also that, for a massless conformally coupled field (
$m=0$
and
$\Delta \xi =0$
), the second term on the righthand side of Equation ( 123 ) will not contribute in the correlation functions ( 120 ), since in this case the stochastic source is traceless.
A detailed analysis given in [
209]
concludes that the homogeneous solution
${\stackrel{~}{G}}_{h}^{\left(1\right)\mu \nu}\left(p\right)$
gives no contribution to the correlation functions ( 120 ). Consequently
${\mathcal{G}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})=\langle {G}_{p}^{\left(1\right)\mu \nu}\left(x\right){G}_{p}^{\left(1\right)\alpha \beta}\left({x}^{\prime}\right){\rangle}_{s}$
, where
${G}_{p}^{\left(1\right)\mu \nu}\left(x\right)$
is the inverse Fourier transform of Equation ( 126 ), and the correlation functions ( 120 ) are
$$\begin{array}{c}{\langle {\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right){\stackrel{~}{G}}_{p}^{\left(1\right)\alpha \beta}\left({p}^{\prime}\right)\rangle}_{s}=64\left(2\pi {)}^{6}{G}^{2}{\delta}^{4}\right(p+{p}^{\prime}\left){{D}_{p}^{\mu \nu}}_{\rho \sigma}\right(p\left){{D}_{p}^{\alpha \beta}}_{\lambda \gamma}\right(p\left){\stackrel{~}{N}}^{\rho \sigma \lambda \gamma}\right(p).\end{array}$$ 
(127)

It is easy to see from the above analysis that the prescriptions
$[{]}_{r}$
in the factors
${D}_{p}$
are irrelevant in the last expression and, thus, they can be suppressed. Taking into account that
${F}_{l}(p)={F}_{l}^{*}\left(p\right)$
, with
$l=1,2,3$
, we get from Equations ( 123 ) and ( 124 )
$$\begin{array}{ccc}{\langle {\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right){\stackrel{~}{G}}_{p}^{\left(1\right)\alpha \beta}\left({p}^{\prime}\right)\rangle}_{s}& =& 64(2\pi {)}^{6}{G}^{2}\frac{{\delta}^{4}(p+{p}^{\prime})}{{\left{F}_{1}\left(p\right)\right}^{2}}[{\stackrel{~}{N}}^{\mu \nu \alpha \beta}\left(p\right)\frac{{F}_{2}\left(p\right)}{{F}_{3}\left(p\right)}{p}^{2}{P}^{\mu \nu}{\stackrel{~}{N}}_{\rho}^{\alpha \beta \rho}\left(p\right)\end{array}$$  
$$\begin{array}{ccc}& & \frac{{F}_{2}^{*}\left(p\right)}{{F}_{3}^{*}\left(p\right)}{p}^{2}{P}^{\alpha \beta}{\stackrel{~}{N}}_{\rho}^{\mu \nu \rho}\left(p\right)+\frac{{\left{F}_{2}\left(p\right)\right}^{2}}{{\left{F}_{3}\left(p\right)\right}^{2}}{p}^{2}{P}^{\mu \nu}{p}^{2}{P}^{\alpha \beta}{{\stackrel{~}{N}}_{\rho}^{\rho}}_{\sigma}^{\sigma}\left(p\right)].\end{array}$$ 
(128)

This last expression is well defined as a bidistribution and can be easily evaluated using Equation ( 118 ).
The final explicit result for the Fourier transformed correlation function for the Einstein tensor is thus
$$\begin{array}{ccc}{\langle {\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right){\stackrel{~}{G}}_{p}^{\left(1\right)\alpha \beta}\left({p}^{\prime}\right)\rangle}_{s}& =& \frac{2}{45}\left(2\pi {)}^{5}{G}^{2}\frac{{\delta}^{4}(p+{p}^{\prime})}{{\left{F}_{1}\left(p\right)\right}^{2}}\theta \right({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}\end{array}$$  
$$\begin{array}{ccc}& & \times [\frac{1}{4}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}({p}^{2}{)}^{2}{P}^{\mu \nu \alpha \beta}\end{array}$$  
$$\begin{array}{ccc}& & +10{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}({p}^{2}{)}^{2}{P}^{\mu \nu}{P}^{\alpha \beta}{\left13{p}^{2}\frac{{F}_{2}\left(p\right)}{{F}_{3}\left(p\right)}\right}^{2}].\end{array}$$ 
(129)

To obtain the correlation functions in coordinate space, Equation ( 120 ), we take the inverse Fourier transform. The final result is
$$\begin{array}{c}{\mathcal{G}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})=\frac{\pi}{45}{G}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}{\mathcal{G}}_{A}(x{x}^{\prime})+\frac{8\pi}{9}{G}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}{\mathcal{G}}_{B}(x{x}^{\prime}),\end{array}$$ 
(130)

with
$$\begin{array}{c}\begin{array}{ccc}{\stackrel{~}{\mathcal{G}}}_{A}\left(p\right)& \equiv & \theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}{\left(1+4\frac{{m}^{2}}{{p}^{2}}\right)}^{2}\frac{1}{{\left{F}_{1}\left(p\right)\right}^{2}},\\ {\stackrel{~}{\mathcal{G}}}_{B}\left(p\right)& \equiv & \theta ({p}^{2}4{m}^{2})\sqrt{1+4\frac{{m}^{2}}{{p}^{2}}}{\left(3\Delta \xi +\frac{{m}^{2}}{{p}^{2}}\right)}^{2}\frac{1}{{\left{F}_{1}\left(p\right)\right}^{2}}{\left13{p}^{2}\frac{{F}_{2}\left(p\right)}{{F}_{3}\left(p\right)}\right}^{2},\end{array}\end{array}$$ 
(131)

where
${F}_{l}\left(p\right)$
,
$l=1,2,3$
, are given in Equations ( 116 ) and ( 125 ). Notice that, for a massless field (
$m=0$
), we have
$$\begin{array}{c}\begin{array}{ccc}{F}_{1}\left(p\right)& =& 1+4\pi G{p}^{2}\stackrel{~}{H}(p;{\overline{\mu}}^{2}),\\ {F}_{2}\left(p\right)& =& \frac{16}{3}\pi G\left[(1+180\Delta {\xi}^{2})\frac{1}{4}\stackrel{~}{H}(p;{\overline{\mu}}^{2})6\Upsilon \right],\\ {F}_{3}\left(p\right)& =& 148\pi G{p}^{2}\left[15\Delta {\xi}^{2}\stackrel{~}{H}(p;{\overline{\mu}}^{2})2\Upsilon \right],\end{array}\end{array}$$ 
(132)

with
$\overline{\mu}\equiv \mu exp\left(1920{\pi}^{2}\overline{\alpha}\right)$
and
$\Upsilon \equiv \overline{\beta}60\Delta {\xi}^{2}\overline{\alpha}$
, and where
$\stackrel{~}{H}(p;{\mu}^{2})$
is the Fourier transform of
$H(x;{\mu}^{2})$
given in Equation ( 110 ).
6.4.2 Correlation functions for the metric perturbations
Starting from the solutions found for the linearized Einstein tensor, which are characterized by the twopoint correlation functions ( 130 ) (or, in terms of Fourier transforms, Equation ( 129 )), we can now solve the equations for the metric perturbations. Working in the harmonic gauge,
${\partial}_{\nu}{\overline{h}}^{\mu \nu}=0$
(this zero must be understood in a statistical sense) where
${\overline{h}}_{\mu \nu}\equiv {h}_{\mu \nu}\frac{1}{2}{\eta}_{\mu \nu}{h}_{\alpha}^{\alpha}$
, the equations for the metric perturbations in terms of the Einstein tensor are
$$\begin{array}{c}\square {\overline{h}}^{\mu \nu}\left(x\right)=2{G}^{\left(1\right)\mu \nu}\left(x\right),\end{array}$$ 
(133)

or, in terms of Fourier transforms,
${p}^{2}{\stackrel{~}{\overline{h}}}^{\mu \nu}\left(p\right)=2{\stackrel{~}{G}}^{\left(1\right)\mu \nu}\left(p\right)$
. Similarly to the analysis of the equation for the Einstein tensor, we can write
${\overline{h}}^{\mu \nu}=\langle {\overline{h}}^{\mu \nu}{\rangle}_{s}+{\overline{h}}_{f}^{\mu \nu}$
, where
${\overline{h}}_{f}^{\mu \nu}$
is a solution to these equations with zero average, and the twopoint correlation functions are defined by
$$\begin{array}{ccc}{\mathcal{\mathscr{H}}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})& \equiv & {\langle {\overline{h}}^{\mu \nu}\left(x\right){\overline{h}}^{\alpha \beta}\left({x}^{\prime}\right)\rangle}_{s}{\langle {\overline{h}}^{\mu \nu}\left(x\right)\rangle}_{s}{\langle {\overline{h}}^{\alpha \beta}\left({x}^{\prime}\right)\rangle}_{s}\end{array}$$  
$$\begin{array}{ccc}& =& {\langle {\overline{h}}_{f}^{\mu \nu}\left(x\right){\overline{h}}_{f}^{\alpha \beta}\left({x}^{\prime}\right)\rangle}_{s}.\end{array}$$ 
(134)

We can now seek solutions of the Fourier transform of Equation ( 133 ) of the form
${\stackrel{~}{\overline{h}}}_{f}^{\mu \nu}\left(p\right)=2D\left(p\right){\stackrel{~}{G}}_{f}^{\left(1\right)\mu \nu}\left(p\right)$
, where
$D\left(p\right)$
is a Lorentz invariant scalar distribution in Minkowski spacetime, which is the most general solution of
${p}^{2}D\left(p\right)=1$
. Note that, since the linearized Einstein tensor is conserved, solutions of this form automatically satisfy the harmonic gauge condition. As in the previous subsection, we can write
$D\left(p\right)=[1/{p}^{2}{]}_{r}+{D}_{h}(p)$
, where
${D}_{h}\left(p\right)$
is the most general solution to the associated homogeneous equation and, correspondingly, we have
${\stackrel{~}{\overline{h}}}_{f}^{\mu \nu}\left(p\right)={\stackrel{~}{\overline{h}}}_{p}^{\mu \nu}\left(p\right)+{\stackrel{~}{\overline{h}}}_{h}^{\mu \nu}\left(p\right)$
.
However, since
${D}_{h}\left(p\right)$
has support on the set of points for which
${p}^{2}=0$
, it is easy to see from Equation ( 129 ) (from the factor
$\theta ({p}^{2}4{m}^{2})$
) that
$\langle {\stackrel{~}{\overline{h}}}_{h}^{\mu \nu}\left(p\right){\stackrel{~}{G}}_{f}^{\left(1\right)\alpha \beta}\left({p}^{\prime}\right){\rangle}_{s}=0$
and, thus, the twopoint correlation functions ( 134 ) can be computed from
$\langle {\stackrel{~}{\overline{h}}}_{f}^{\mu \nu}\left(p\right){\stackrel{~}{\overline{h}}}_{f}^{\alpha \beta}\left({p}^{\prime}\right){\rangle}_{s}=\langle {\stackrel{~}{\overline{h}}}_{p}^{\mu \nu}\left(p\right){\stackrel{~}{\overline{h}}}_{p}^{\alpha \beta}\left({p}^{\prime}\right){\rangle}_{s}$
. From Equation ( 129 ) and due to the factor
$\theta ({p}^{2}4{m}^{2})$
, it is also easy to see that the prescription
$[{]}_{r}$
is irrelevant in this correlation function, and we obtain
$$\begin{array}{c}{\langle {\stackrel{~}{\overline{h}}}_{p}^{\mu \nu}\left(p\right){\stackrel{~}{\overline{h}}}_{p}^{\alpha \beta}\left({p}^{\prime}\right)\rangle}_{s}=\frac{4}{({p}^{2}{)}^{2}}{\langle {\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right){\stackrel{~}{G}}_{p}^{\left(1\right)\alpha \beta}\left({p}^{\prime}\right)\rangle}_{s},\end{array}$$ 
(135)

where
$\langle {\stackrel{~}{G}}_{p}^{\left(1\right)\mu \nu}\left(p\right){\stackrel{~}{G}}_{p}^{\left(1\right)\alpha \beta}\left({p}^{\prime}\right){\rangle}_{s}$
is given by Equation ( 129 ). The righthand side of this equation is a well defined bidistribution, at least for
$m\ne 0$
(the
$\theta $
function provides the suitable cutoff ). In the massless field case, since the noise kernel is obtained as the limit
$m\to 0$
of the noise kernel for a massive field, it seems that the natural prescription to avoid the divergences on the lightcone
${p}^{2}=0$
is a Hadamard finite part (see [
256,
302]
for its definition). Taking this prescription, we also get a well defined bidistribution for the massless limit of the last expression.
The final result for the twopoint correlation function for the field
${\overline{h}}^{\mu \nu}$
is
$$\begin{array}{c}{\mathcal{\mathscr{H}}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})=\frac{4\pi}{45}{G}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}{\mathcal{\mathscr{H}}}_{A}(x{x}^{\prime})+\frac{32\pi}{9}{G}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu}{\mathcal{\mathcal{F}}}_{x}^{\alpha \beta}{\mathcal{\mathscr{H}}}_{B}(x{x}^{\prime}),\end{array}$$ 
(136)

where
${\stackrel{~}{\mathcal{\mathscr{H}}}}_{A}\left(p\right)\equiv [1/({p}^{2}{)}^{2}\left]{\stackrel{~}{\mathcal{G}}}_{A}\right(p)$
and
${\stackrel{~}{\mathcal{\mathscr{H}}}}_{B}\left(p\right)\equiv [1/({p}^{2}{)}^{2}\left]{\stackrel{~}{\mathcal{G}}}_{B}\right(p)$
, with
${\stackrel{~}{\mathcal{G}}}_{A}\left(p\right)$
and
${\stackrel{~}{\mathcal{G}}}_{B}\left(p\right)$
given by Equation ( 131 ). The twopoint correlation functions for the metric perturbations can be easily obtained using
${h}_{\mu \nu}={\overline{h}}_{\mu \nu}\frac{1}{2}{\eta}_{\mu \nu}{\overline{h}}_{\alpha}^{\alpha}$
.
6.4.3 Conformally coupled field
For a conformally coupled field, i.e., when
$m=0$
and
$\Delta \xi =0$
, the previous correlation functions are greatly simplified and can be approximated explicitly in terms of analytic functions. The detailed results are given in [
209]
; here we outline the main features.
When
$m=0$
and
$\Delta \xi =0$
, we have
${\mathcal{G}}_{B}\left(x\right)=0$
and
${\stackrel{~}{\mathcal{G}}}_{A}\left(p\right)=\theta ({p}^{2}){\left{F}_{1}\left(p\right)\right}^{2}$
. Thus the twopoint correlations functions for the Einstein tensor is
$$\begin{array}{c}{\mathcal{G}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})=\frac{\pi}{45}{G}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}\frac{{e}^{ip(x{x}^{\prime})}\theta ({p}^{2})}{{\left1+4\pi G{p}^{2}\stackrel{~}{H}(p;{\overline{\mu}}^{2})\right}^{2}},\end{array}$$ 
(137)

where
$\stackrel{~}{H}(p,{\mu}^{2})=(480{\pi}^{2}{)}^{1}ln[\left(\right({p}^{0}+i\epsilon {)}^{2}+{p}^{i}{p}_{i})/{\mu}^{2}]$
(see Equation ( 110 )).
To estimate this integral, let us consider spacelike separated points
$(x{x}^{\prime}{)}^{\mu}=(0,\mathbf{x}{\mathbf{x}}^{\prime})$
, and define
$\mathbf{y}=\mathbf{x}{\mathbf{x}}^{\prime}$
. We may now formally change the momentum variable
${p}^{\mu}$
by the dimensionless vector
${s}^{\mu}$
,
${p}^{\mu}={s}^{\mu}/\mathbf{y}$
. Then the previous integral denominator is
$1+16\pi ({L}_{P}/\mathbf{y}{)}^{2}{s}^{2}\stackrel{~}{H}\left(s\right){}^{2}$
, where we have introduced the Planck length
${L}_{P}=\sqrt{G}$
. It is clear that we can consider two regimes:
(a) when
${L}_{P}\ll \mathbf{y}$
, and (b) when
$\mathbf{y}\sim {L}_{P}$
. In case (a) the correlation function, for the
$0000$
component, say, will be of the order
$${\mathcal{G}}^{0000}(\mathbf{y})\sim \frac{{L}_{P}^{4}}{\mathbf{y}{}^{8}}.$$
In case (b) when the denominator has zeros, a detailed calculation carried out in [
209]
shows that
$${\mathcal{G}}^{0000}(\mathbf{y})\sim {e}^{\mathbf{y}/{L}_{P}}\left(\frac{{L}_{P}}{\mathbf{y}{}^{5}}+\cdot \cdot \cdot +\frac{1}{{L}_{P}^{2}\mathbf{y}{}^{2}}\right),$$
which indicates an exponential decay at distances around the Planck scale. Thus short scale fluctuations are strongly suppressed.
For the twopoint metric correlation the results are similar. In this case we have
$$\begin{array}{c}{\mathcal{\mathscr{H}}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})=\frac{4\pi}{45}{G}^{2}{\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}\frac{{e}^{ip(x{x}^{\prime})}\theta ({p}^{2})}{({p}^{2}{)}^{2}{\left1+4\pi G{p}^{2}\stackrel{~}{H}(p;{\overline{\mu}}^{2})\right}^{2}}.\end{array}$$ 
(138)

The integrand has the same behavior of the correlation function of Equation ( 137 ), thus matter fields tends to suppress the short scale metric perturbations. In this case we find, as for the correlation of the Einstein tensor, that for case (a) above we have
$${\mathcal{\mathscr{H}}}^{0000}(\mathbf{y})\sim \frac{{L}_{P}^{4}}{\mathbf{y}{}^{4}},$$
and for case (b) we have
$${\mathcal{\mathscr{H}}}^{0000}(\mathbf{y})\sim {e}^{\mathbf{y}/{L}_{P}}\left(\frac{{L}_{P}}{\mathbf{y}}+...\right).$$
It is interesting to write expression ( 138 ) in an alternative way. If we use the dimensionless tensor
${P}^{\mu \nu \alpha \beta}$
introduced in Equation ( 100 ), which accounts for the effect of the operator
${\mathcal{\mathcal{F}}}_{x}^{\mu \nu \alpha \beta}$
, we can write
$$\begin{array}{c}{\mathcal{\mathscr{H}}}^{\mu \nu \alpha \beta}(x,{x}^{\prime})=\frac{4\pi}{45}{G}^{2}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}\frac{{e}^{ip(x{x}^{\prime})}{P}^{\mu \nu \alpha \beta}\theta ({p}^{2})}{{\left1+4\pi G{p}^{2}\stackrel{~}{H}(p;{\overline{\mu}}^{2})\right}^{2}}.\end{array}$$ 
(139)

This expression allows a direct comparison with the graviton propagator for linearized quantum gravity in the
$1/N$
expansion found by Tomboulis [
277]
. One can see that the imaginary part of the graviton propagator leads, in fact, to Equation ( 139 ). In [
255]
it is shown that the twopoint correlation functions for the metric perturbations derived from the Einstein–Langevin equation are equivalent to the symmetrized quantum twopoint correlation functions for the metric fluctuations in the large
$N$
expansion of quantum gravity interacting with
$N$
matter fields.
6.5 Discussion
The main results of this section are the correlation functions ( 130 ) and ( 136 ). In the case of a conformal field, the correlation functions of the linearized Einstein tensor have been explicitly estimated. From the exponential factors
${e}^{\mathbf{y}/{L}_{P}}$
in these results for scales near the Planck length, we see that the correlation functions of the linearized Einstein tensor have the Planck length as the correlation length. A similar behavior is found for the correlation functions of the metric perturbations. Since these fluctuations are induced by the matter fluctuations, we infer that the effect of the matter fields is to suppress the fluctuations of the metric at very small scales. On the other hand, at scales much larger than the Planck length, the induced metric fluctuations are small compared with the free graviton propagator which goes like
${L}_{P}^{2}/\mathbf{y}{}^{2}$
, since the action for the free graviton goes like
${S}_{h}\sim \int {d}^{4}x{L}_{P}^{2}h\square h$
.
For background solutions of semiclassical gravity with other scales present apart from the Planck scales (for instance, for matter fields in a thermal state), stressenergy fluctuations may be important at larger scales. For such backgrounds, stochastic semiclassical gravity might predict correlation functions with characteristic correlation lengths larger than the Planck scales. It seems quite plausible, nevertheless, that these correlation functions would remain nonanalytic in their characteristic correlation lengths. This would imply that these correlation functions could not be obtained from a calculation involving a perturbative expansion in the characteristic correlation lengths. In particular, if these correlation lengths are proportional to the Planck constant
$\u0127$
, the gravitational correlation functions could not be obtained from an expansion in
$\u0127$
. Hence, stochastic semiclassical gravity might predict a behavior for gravitational correlation functions different from that of the analogous functions in perturbative quantum gravity [
78,
77,
79,
80]
. This is not necessarily inconsistent with having neglected action terms of higher order in
$\u0127$
when considering semiclassical gravity as an effective theory [
91]
. It is, in fact, consistent with the closed connection of stochastic gravity with the large
$N$
expansion of quantum gravity interacting with
$N$
matter fields.
7 Structure Formation
Cosmological structure formation is a key problem in modern cosmology [
190,
229]
and inflation offers a natural solution to this problem. If an inflationary period is present, the initial seeds for the generation of the primordial inhomogeneities that lead to the large scale structure have their source in the quantum fluctuations of the inflaton field, the field which is generally responsible for driving inflation. Stochastic gravity provides a sound and natural formalism for the derivation of the cosmological perturbations generated during inflation.
In [
254]
it was shown that the correlation functions that follow from the Einstein–Langevin equation which emerges in the framework of stochastic gravity coincide with that obtained with the usual quantization procedures [
218]
, when both the metric perturbations and the inflaton fluctuations are both linearized. Stochastic gravity, however, can naturally deal with the fluctuations of the inflaton field even beyond the linear approximation.
Here we will illustrate the equivalence with the usual formalism, based on the quantization of the linear cosmological and inflaton perturbations, with one of the simplest chaotic inflationary models in which the background spacetime is a quaside Sitter universe [
253,
254]
.
7.1 The model
In this chaotic inflationary model [
199]
the inflaton field
$\phi $
of mass
$m$
is described by the following Lagrangian density:
$$\begin{array}{c}\mathcal{\mathcal{L}}\left(\phi \right)=\frac{1}{2}{g}^{ab}{\nabla}_{a}\phi {\nabla}_{b}\phi +\frac{1}{2}{m}^{2}{\phi}^{2}.\end{array}$$ 
(140)

The conditions for the existence of an inflationary period, which is characterized by an accelerated cosmological expansion, is that the value of the field over a region with the typical size of the Hubble radius is higher than the Planck mass
${m}_{P}$
. This is because in order to solve the cosmological horizon and flatness problem more than 60 efolds of expansion are needed; to achieve this the scalar field should begin with a value higher than
$3{m}_{P}$
. The inflaton mass is small: As we will see, the large scale anisotropies measured in the cosmic background radiation [
265]
restrict the inflaton mass to be of the order of
${10}^{6}{m}_{P}$
. We will not discuss the naturalness of this inflationary model and we will simply assume that if one such region is found (inside a much larger universe) it will inflate to become our observable universe.
We want to study the metric perturbations produced by the stressenergy tensor fluctuations of the inflaton field on the homogeneous background of a flat Friedmann–Robertson–Walker model, described by the cosmological scale factor
$a\left(\eta \right)$
, where
$\eta $
is the conformal time, which is driven by the homogeneous inflaton field
$\phi \left(\eta \right)=\langle \hat{\phi}\rangle $
. Thus we write the inflaton field in the following form:
$$\begin{array}{c}\hat{\phi}=\phi \left(\eta \right)+\hat{\phi}\left(x\right),\end{array}$$ 
(141)

where
$\hat{\phi}\left(x\right)$
corresponds to a free massive quantum scalar field with zero expectation value on the homogeneous background metric,
$\langle \hat{\phi}\rangle =0$
. We will restrict ourselves to scalartype metric perturbations, because these are the ones that couple to the inflaton fluctuations in the linear theory. We note that this is not so if we were to consider inflaton fluctuations beyond the linear approximation; then tensorial and vectorial metric perturbations would also be driven. The perturbed metric
${\stackrel{~}{g}}_{ab}={g}_{ab}+{h}_{ab}$
can be written in the longitudinal gauge as
$$\begin{array}{c}d{s}^{2}={a}^{2}\left(\eta \right)\left[(1+2\Phi (x\left)\right)d{\eta}^{2}+(12\Psi (x\left)\right){\delta}_{ij}d{x}^{i}d{x}^{j}\right],\end{array}$$ 
(142)

where the scalar metric perturbations
$\Phi \left(x\right)$
and
$\Psi \left(x\right)$
correspond to Bardeen's gauge invariant variables [
12]
.
7.2 The Einstein–Langevin equation for scalar metric perturbations
The Einstein–Langevin equation as described in Section 3 is gauge invariant, and thus we can work in a desired gauge and then extract the gauge invariant quantities. The Einstein–Langevin equation ( 14 ) reads now
$$\begin{array}{c}{G}_{ab}^{\left(0\right)}8\pi G\langle {\hat{T}}_{ab}^{\left(0\right)}\rangle +{G}_{ab}^{\left(1\right)}\left(h\right)8\pi G\langle {\hat{T}}_{ab}^{\left(1\right)}\left(h\right)\rangle =8\pi G{\xi}_{ab},\end{array}$$ 
(143)

where the two first terms cancel, that is
${G}_{ab}^{\left(0\right)}8\pi G\langle {\hat{T}}_{ab}^{\left(0\right)}\rangle =0$
, as the background metric satisfies the semiclassical Einstein equations. Here the superscripts
${}^{\left(0\right)}$
and
${}^{\left(1\right)}$
refer to functions in the background metric
${g}_{ab}$
and linear in the metric perturbation
${h}_{ab}$
, respectively. The stress tensor operator
${\hat{T}}_{ab}$
for the minimally coupled inflaton field in the perturbed metric is
$$\begin{array}{c}{\hat{T}}_{ab}={\stackrel{~}{\nabla}}_{a}\hat{\phi}{\stackrel{~}{\nabla}}_{b}\hat{\phi}+\frac{1}{2}{\stackrel{~}{g}}_{ab}\left({\stackrel{~}{\nabla}}_{c}\hat{\phi}{\stackrel{~}{\nabla}}^{c}\hat{\phi}+{m}^{2}{\hat{\phi}}^{2}\right).\end{array}$$ 
(144)

Using the decomposition of the scalar field into its homogeneous and inhomogeneous part, see Equation ( 141 ), and the metric
${\stackrel{~}{g}}_{ab}$
into its homogeneous background
${g}_{ab}$
and its perturbation
${h}_{ab}$
, the renormalized expectation value for the stressenergy tensor operator can be written as
$$\begin{array}{c}\langle {\hat{T}}_{ab}^{R}\left[\stackrel{~}{g}\right]\rangle ={\langle {\hat{T}}_{ab}\left[\stackrel{~}{g}\right]\rangle}_{\phi \phi}+{\langle {\hat{T}}_{ab}\left[\stackrel{~}{g}\right]\rangle}_{\phi \phi}+{\langle {\hat{T}}_{ab}^{R}\left[\stackrel{~}{g}\right]\rangle}_{\phi \phi},\end{array}$$ 
(145)

where the subindices indicate the degree of dependence on the homogeneous field
$\phi $
, and its perturbation
$\phi $
. The first term in this equation depends only on the homogeneous field and it is given by the classical expression. The second term is proportional to
$\langle \hat{\phi}\left[\stackrel{~}{g}\right]\rangle $
which is not zero because the field dynamics is considered on the perturbed spacetime, i.e., this term includes the coupling of the field with
${h}_{ab}$
and may be obtained from the expectation value of the linearized Klein–Gordon equation,
$$\begin{array}{c}\left({\square}_{g+h}{m}^{2}\right)\hat{\phi}=0.\end{array}$$ 
(146)

The last term in Equation ( 145 ) corresponds to the expectation value to the stress tensor for a free scalar field on the spacetime of the perturbed metric.
After using the previous decomposition, the noise kernel
${N}_{abcd}[g;x,y)$
defined in Equation ( 11 ) can be written as
$$\begin{array}{c}\langle \left\{{\hat{t}}_{ab}[g;x),{\hat{t}}_{cd}[g;y)\right\}\rangle ={\langle \left\{{\hat{t}}_{ab}[g;x),{\hat{t}}_{cd}[g;y)\right\}\rangle}_{(\phi \phi {)}^{2}}+{\langle \left\{{\hat{t}}_{ab}[g;x),{\hat{t}}_{cd}[g;y)\right\}\rangle}_{(\phi \phi {)}^{2}},\end{array}$$ 
(147)

where we have used the fact that
$\langle \hat{\phi}\rangle =0=\langle \hat{\phi}\hat{\phi}\hat{\phi}\rangle $
for Gaussian states on the background geometry. We consider the vacuum state to be the Euclidean vacuum which is preferred in the de Sitter background, and this state is Gaussian. In the above equation the first term is quadratic in
$\hat{\phi}$
, whereas the second one is quartic. Both contributions to the noise kernel are separately conserved since both
$\phi \left(\eta \right)$
and
$\hat{\phi}$
satisfy the Klein–Gordon field equations on the background spacetime.
Consequently, the two terms can be considered separately. On the other hand, if one treats
$\hat{\phi}$
as a small perturbation, the second term in Equation ( 147 ) is of lower order than the first and may be consistently neglected; this corresponds to neglecting the last term of Equation ( 145 ). The stress tensor fluctuations due to a term of that kind were considered in [
252]
.
We can now write down the Einstein–Langevin equations (
143 ) to linear order in the inflaton fluctuations. It is easy to check [
254]
that the spacespace components coming from the stress tensor expectation value terms and the stochastic tensor are diagonal, i.e.,
$\langle {\hat{T}}_{ij}\rangle =0={\xi}_{ij}$
for
$i\ne j$
. This, in turn, implies that the two functions characterizing the scalar metric perturbations are equal,
$\Phi =\Psi $
, in agreement with [
218]
. The equation for
$\Phi $
can be obtained from the
$0i$
component of the Einstein–Langevin equation, which in Fourier space reads