Once the fluctuations of the stress-energy operator have been characterized, we can perturbatively extend the semiclassical theory to account for such fluctuations. Thus we will assume that the background spacetime metric is a solution of the semiclassical Einstein equations (8) and we will write the new metric for the extended theory as , where we will assume that is a perturbation to the background solution. The renormalized stress-energy operator and the state of the quantum field may now be denoted by and , respectively, and will be the corresponding expectation value.

Let us now introduce a Gaussian stochastic tensor field defined by the following correlators:

where means statistical average. The symmetry and positive semi-definite property of the noise kernel guarantees that the stochastic field tensor , or for short, just introduced is well-defined. Note that this stochastic tensor captures only partially the quantum nature of the fluctuations of the stress-energy operator as it assumes that cumulants of higher order are zero.An important property of this stochastic tensor is that it is covariantly conserved in the background spacetime . In fact, as a consequence of the conservation of one can see that . Taking the divergence in Equation (14) one can then show that and so that is deterministic and represents with certainty the zero vector field in .

For a conformal field, i.e., a field whose classical action is conformally invariant, is traceless: ; so that, for a conformal matter field the stochastic source gives no correction to the trace anomaly. In fact, from the trace anomaly result, which states that is, in this case, a local c-number functional of times the identity operator, we have that . It then follows from Equation (14) that and ; an alternative proof based on the point-separation method is given in [304, 305], see also Section 5.

All these properties make it quite natural to incorporate into the Einstein equations the stress-energy fluctuations by using the stochastic tensor as the source of the metric perturbations. Thus we will write the following equation:

This equation is in the form of a (semiclassical) Einstein–Langevin equation; it is a dynamical equation for the metric perturbation to linear order. It describes the backreaction of the metric to the quantum fluctuations of the stress-energy tensor of matter fields, and gives a first-order extension to semiclassical gravity as described by the semiclassical Einstein equation (8).Note that we refer to the Einstein–Langevin equation as a first-order extension to the semiclassical Einstein equation of semiclassical gravity and the lowest-level representation of stochastic gravity. However, stochastic gravity has a much broader meaning; it refers to the range of theories based on second and higher-order correlation functions. Noise can be defined in effectively-open systems (e.g., correlation noise [61] in the Schwinger–Dyson equation hierarchy) to some degree, but one should not expect the Langevin form to prevail. In this sense we say stochastic gravity is the intermediate theory between semiclassical gravity (a mean field theory based on the expectation values of the energy momentum tensor of quantum fields) and quantum gravity (the full hierarchy of correlation functions retaining complete quantum coherence [187, 188].

The renormalization of the operator is carried out exactly as in the previous case, now in the perturbed metric . Note that the stochastic source is not dynamical; it is independent of , since it describes the fluctuations of the stress tensor on the semiclassical background .

An important property of the Einstein–Langevin equation is that it is gauge invariant under the change of by , where is a stochastic vector field on the background manifold . Note that a tensor such as transforms as to linear order in the perturbations, where is the Lie derivative with respect to . Now, let us write the source tensors in Equations (15) and (8) to the left-hand sides of these equations. If we substitute with in this new version of Equation (15), we get the same expression, with instead of , plus the Lie derivative of the combination of tensors, which appear on the left-hand side of the new Equation (8). This last combination vanishes when Equation (8) is satisfied, i.e., when the background metric is a solution of semiclassical gravity.

From the statistical average of Equation (15) we have that must be a solution of the semiclassical Einstein equation linearized around the background ; this solution has been proposed as a test for the validity of the semiclassical approximation [10, 11] a point that will be further discussed in Section 3.3.

The stochastic equation (15) predicts that the gravitational field has stochastic fluctuations over the background . This equation is linear in , thus its solutions can be written as follows,

where is the solution of the homogeneous equation containing information on the initial conditions and is the retarded propagator of Equation (15) with vanishing initial conditions. Form this we obtain the two-point correlation functions for the metric perturbations: There are two different contributions to the two-point correlations, which we have distinguished in the second equality. The first one is connected to the fluctuations of the initial state of the metric perturbations and we will refer to them as intrinsic fluctuations. The second contribution is proportional to the noise kernel and is thus connected with the fluctuations of the quantum fields; we will refer to them as induced fluctuations. To find these two-point stochastic correlation functions one needs to know the noise kernel . Explicit expressions of this kernel in terms of the two-point Wightman functions is given in [258], expressions based on point-splitting methods have also been given in [304, 315]. Note that the noise kernel should be thought of as a distribution function, the limit of coincidence points has meaning only in the sense of distributions.The two-point stochastic correlation functions for the metric perturbations of Equation (17) satisfy a very important property. In fact, it can be shown that they correspond exactly to the symmetrized two-point correlation functions for the quantum metric perturbations in the large expansion, i.e., the quantum theory describing the interaction of the gravitational field with arbitrary free fields and expanded in powers of . To leading order for the graviton propagator one finds that

where is the quantum operator corresponding to the metric perturbations and the statistical average in Equation (17) for the homogeneous solutions is now taken with respect to the Wigner distribution that describes the initial quantum state of the metric perturbations. The Lorentz gauge condition , as well as an initial condition to completely fix the gauge of the initial state, should be implicitly understood. Moreover, since there are now scalar fields, the stochastic source has been rescaled so that the two-point correlation defined by Equation (14) should be times the noise kernel of a single field. This result was implicitly obtained in the Minkowski background in [259] where the two-point correlation in the stochastic context was computed for the linearized metric perturbations. This stochastic correlation exactly agrees with the symmetrized part of the graviton propagator computed by Tomboulis [348] in the quantum context of gravity interacting with Fermion fields, where the graviton propagator is of order . This result can be extended to an arbitrary background in the context of the large expansion; a sketch of the proof with explicit details in the Minkowski background can be found in [203]. This connection between the stochastic correlations and the quantum correlations was noted and studied in detail in the context of simpler open quantum systems [66]. Stochastic gravity goes beyond semiclassical gravity in the following sense. The semiclassical theory, which is based on the expectation value of the stress-energy tensor, carries information on the field two-point correlations only, since is quadratic in the field operator . The stochastic theory, on the other hand, is based on the noise kernel 12, which is quartic in the field operator. However, it does not carry information on the graviton-graviton interaction, which in the context of the large expansion gives diagrams of order . This will be illustrated in Section 3.3.1. Furthermore, the retarded propagator also gives information on the commutator so that combining the commutator with the anticommutator, the quantum two-point correlation functions are determined. Moreover, assuming a Gaussian initial state with vanishing expectation value for the metric perturbations, any -point quantum correlation function is determined by the two-point quantum correlations and thus by the stochastic approach. Consequently, one may regard the Einstein–Langevin equation as a useful intermediary tool to compute the correlation functions for the quantum metric perturbations.We should, however, also emphasize that Langevin-like equations are obtained to describe the quantum to classical transition in open quantum systems, when quantum decoherence takes place by coarse-graining of the environment as well as by suitable coarse-graining of the system variables [101, 127, 144, 146, 150, 374]. In those cases the stochastic correlation functions describe actual classical correlations of the system variables. Examples can be found in the case of a moving charged particle in an electromagnetic field in quantum electrodynamics [219] and in several quantum Brownian models [64, 65, 66].

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