Stochastic gravity

3.2 Stochastic gravity

The purpose of stochastic gravity is to extend the semiclassical theory to account for these fluctuations in a self-consistent way. A physical observable that describes these fluctuations to lowest order is the noise kernel bi-tensor, which is defined through the two point correlation of the stress-energy operator as

where the curly brackets mean anticommutator, and where

This bi-tensor can also be written as $Nab,c′d′[g;x,y)$ , or $Nab,c′d′(x,y)$ 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 stress-tensor operator $Tˆab[g;x)$ on a given background metric $gab$ , thus a regulator is implicitly assumed on the right-hand side of Equation (11

). However, for a linear quantum field the above kernel – the expectation function of a bi-tensor – is free of ultraviolet divergences because the regularized $Tab[g;x)$ differs from the renormalized $R T ab[g;x)$ 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 $⟨ˆT (x )Tˆ (y)⟩ ab cd$ is the same as that of $ˆ ˆ ⟨Tab(x )⟩⟨Tcd(y)⟩$ , and $ˆ Tab$ can be replaced by the renormalized operator $ˆR Tab$ 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 bi-tensor $Nabcd[g;x,y)$ , or $Nabcd(x,y)$ for short, is real and positive semi-definite, as a consequence of $ˆR T ab$ being self-adjoint. A simple proof is given in [169

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 $g ab$ is a solution of the semiclassical Einstein Equations (7 ), and we will write the new metric for the extended theory as $gab + hab$ , where we will assume that $hab$ is a perturbation to the background solution. The renormalized stress-energy operator and the state of the quantum field may now be denoted by $ˆTRab[g + h ]$ and $|ψ [g + h ]⟩$ , respectively, and $⟨TˆRab[g + h]⟩$ will be the corresponding expectation value.

Let us now introduce a Gaussian stochastic tensor field $ξ [g; x) ab$ defined by the following correlators:

where $⟨...⟩s$ means statistical average. The symmetry and positive semi-definite property of the noise kernel guarantees that the stochastic field tensor $ξab[g,x)$ , or $ξab(x)$ 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 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, $∇a ξab[g;x ) = 0$ . In fact, as a consequence of the conservation of $TˆR[g] ab$ one can see that $∇a N (x,y) = 0 x abcd$ . Taking the divergence in Equation (13 ) one can then show that $⟨∇a ξ ⟩ = 0 abs$ and $a ⟨∇ xξab(x)ξcd(y)⟩s = 0$ , so that $a ∇ ξab$ is deterministic and represents with certainty the zero vector field in $ℳ$ .

For a conformal field, i.e., a field whose classical action is conformally invariant, $ξab$ is traceless: $gabξab[g;x ) = 0$ ; 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 $ab ˆ R g Tab[g]$ is, in this case, a local c-number functional of $gab$ times the identity operator, we have that $gab(x )Nabcd[g;x, y) = 0$ . It then follows from Equation (13 ) that $⟨gabξ ⟩ = 0 ab s$ and $⟨gab(x )ξ (x)ξ (y)⟩ = 0 ab cd s$ ; an alternative proof based on the point-separation method is given in [243 , 245 ] (see also Section 5).

All these properties make it quite natural to incorporate into the Einstein equations the stress-energy fluctuations by using the stochastic tensor $ξab[g;x )$ 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 $hab$ 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 (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 energy-momentum tensor of quantum fields) and quantum gravity (the full hierarchy of correlation functions retaining complete quantum coherence [154 , 155 ]).

The renormalization of the operator $ˆT [g + h] ab$ is carried out exactly as in the previous case, now in the perturbed metric $gab + hab$ . Note that the stochastic source $ξab[g;x )$ is not dynamical; it is independent of $hab$ since it describes the fluctuations of the stress tensor on the semiclassical background $gab$ .

An important property of the Einstein–Langevin equation is that it is gauge invariant under the change of $hab$ by $′ hab = hab + ∇a ζb + ∇bζa$ , where $a ζ$ is a stochastic vector field on the background manifold $ℳ$ . Note that a tensor such as $Rab [g + h]$ transforms as $′ Rab[g + h ] = Rab [g + h ] + ℒζRab [g ]$ to linear order in the perturbations, where $ℒ ζ$ is the Lie derivative with respect to $ζa$ . Now, let us write the source tensors in Equations (14 ) and (7 ) to the left-hand sides of these equations. If we substitute $h$ by $h′$ in this new version of Equation (14 ), we get the same expression, with $h$ instead of $′ h$ , plus the Lie derivative of the combination of tensors which appear on the left-hand side of the new Equation (7 ). This last combination vanishes when Equation (7 ) is satisfied, i.e., when the background metric $g ab$ is a solution of semiclassical gravity.

A solution of Equation (14 ) can be formally written as $hab[ξ]$ . This solution is characterized by the whole family of its correlation functions. From the statistical average of this equation we have that $gab + ⟨hab⟩s$ must be a solution of the semiclassical Einstein equation linearized around the background $gab$ ; 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 $hsab[ξ] = hab[ξ] − ⟨hab⟩s$ . Thus the solutions of the Einstein–Langevin equation will provide the two-point metric correlation functions $⟨hsab(x)hscd(y)⟩s$ .

We see that whereas the semiclassical theory depends on the expectation value of the point-defined value of the stress-energy operator, the stochastic theory carries information also on the two point correlation of the stress-energy operator. We should also emphasize that, even if the metric fluctuations appears classical and stochastic, their origin is quantum not only because they are induced by the fluctuations of quantum matter, but also because they are the suitably coarse-grained variables left over from the quantum gravity fluctuations after some mechanism for decoherence and classicalization of the metric field [106 , 126 , 83 , 120 , 122 , 293 ]. One may, in fact, derive the stochastic semiclassical theory from a full quantum theory. This was done via the world-line influence functional method for a moving charged particle in an electromagnetic field in quantum electrodynamics [178 ]. From another viewpoint, quite independent of whether a classicalization mechanism is mandatory or implementable, the Einstein–Langevin equation proves to be a useful tool to compute the symmetrized two point correlations of the quantum metric perturbations [255 ]. This is illustrated in the linear toy model discussed in [169 ], which has features of some quantum Brownian models [51 , 49 , 50 ].