Go to previous page Go up Go to next page

3.2 Stochastic gravity

The purpose of stochastic gravity is to extend semiclassical theory to account for these fluctuations in a self-consistent way. A physical observable that describes these fluctuations to lowest order is the noise kernel bitensor, which is defined through the two-point correlation of the stress-energy operator as
N [g;x,y ) = 1⟨{ˆt [g;x ),ˆt [g; y)}⟩, (12 ) abcd 2 ab cd
where the curly brackets mean anticommutator, and where
ˆ ˆ ˆtab[g; x) ≡ Tab[g;x ) − ⟨Tab[g;x)⟩. (13 )
This bitensor can also be written 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 (12View Equation). However, for a linear quantum field, the above kernel – the expectation function of a bitensor – 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 (7View Equation)) so that in the subtraction (13View Equation) 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 (12View Equation); an alternative proof of this result is given in [304Jump To The Next Citation Point, 305Jump To The Next Citation Point]. 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 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 [208Jump To The Next Citation Point].

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 (8View Equation) 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:

⟨ξ [g;x)⟩ = 0, ⟨ξ [g;x )ξ [g;y)⟩ = N [g;x, y), (14 ) ab s ab cd s abcd
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 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 ∇aξab[g;x) = 0. In fact, as a consequence of the conservation of ˆTR[g] ab one can see that ∇a N (x,y) = 0 x abcd. Taking the divergence in Equation (14View Equation) 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; 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 gab ˆT R[g] ab is, in this case, a local c-number functional of gab times the identity operator, we have that gab(x)N [g;x,y) = 0 abcd. It then follows from Equation (14View Equation) that ⟨gabξ ⟩ = 0 abs and ab ⟨g (x)ξab(x )ξcd(y)⟩s = 0; an alternative proof based on the point-separation method is given in [304Jump To The Next Citation Point, 305Jump To The Next Citation Point], 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:

( ) Gab[g+h ]+ Λ(gab+hab ) − 2(αAab + βBab )[g +h ]= 8πG ⟨ˆTRab[g +h ]⟩+ ξab[g] . (15 )
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 (8View Equation).

Note that we refer to the Einstein–Langevin equation as a first-order extension to the semiclassical Einstein equation of semiclassical gravity and the lowest-level representation of stochastic gravity. However, stochastic gravity has a much broader meaning; 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 [187Jump To The Next Citation Point, 188Jump To The Next Citation Point].

The renormalization of the operator ˆ Tab[g + h] 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 (15View Equation) and (8View Equation) to the left-hand sides of these equations. If we substitute h with ′ h in this new version of Equation (15View Equation), 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 (8View Equation). This last combination vanishes when Equation (8View Equation) is satisfied, i.e., when the background metric gab is a solution of semiclassical gravity.

From the statistical average of Equation (15View 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 [10Jump To The Next Citation Point, 11Jump To The Next Citation Point] a point that will be further discussed in Section 3.3.

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

∫ ∘ ------- hab(x) = h0ab(x) + 8πG d4x′ − g(x′)Greabtcd(x,x′)ξcd(x′), (16 )
where h0 (x) ab is the solution of the homogeneous equation containing information on the initial conditions and Gret (x,x ′) abcd is the retarded propagator of Equation (15View Equation) with vanishing initial conditions. Form this we obtain the two-point correlation functions for the metric perturbations:
⟨hab(x )hcd(y)⟩s = ⟨h0 (x)h0 (y)⟩s + ab ∫cd ∘ ---------- (8πG )2 d4x′d4y′ g(x′)g(y′)Gret (x,x ′)N efgh(x′,y ′)Gret (y,y′) abef cdgh ≡ ⟨hab(x)hcd(y)⟩int + ⟨hab(x)hcd(y)⟩ind. (17 )
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 Nabcd(x, y). Explicit expressions of this kernel in terms of the two-point Wightman functions is given in [258Jump To The Next Citation Point], expressions based on point-splitting methods have also been given in [304Jump To The Next Citation Point, 315Jump To The Next Citation Point]. 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 (17View Equation) 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 N expansion, i.e., the quantum theory describing the interaction of the gravitational field with N arbitrary free fields and expanded in powers of 1∕N. To leading order for the graviton propagator one finds that

⟨{ˆhab(x ),ˆhcd(y )} ⟩ = 2 ⟨hab(x)hcd(y)⟩s, (18 )
where ˆhab(x) is the quantum operator corresponding to the metric perturbations and the statistical average in Equation (17View Equation) 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 ∇a (h − (1 ∕2)η hc) = 0 ab ab c, as well as an initial condition to completely fix the gauge of the initial state, should be implicitly understood. Moreover, since there are now N scalar fields, the stochastic source has been rescaled so that the two-point correlation defined by Equation (14View Equation) should be 1∕N times the noise kernel of a single field. This result was implicitly obtained in the Minkowski background in [259Jump To The Next Citation Point] 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 [348Jump To The Next Citation Point] in the quantum context of gravity interacting with N Fermion fields, where the graviton propagator is of order 1∕N. This result can be extended to an arbitrary background in the context of the large N expansion; a sketch of the proof with explicit details in the Minkowski background can be found in [203Jump To The Next Citation Point]. This connection between the stochastic correlations and the quantum correlations was noted and studied in detail in the context of simpler open quantum systems [66Jump To The Next Citation Point]. 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 ⟨Tˆab⟩ is quadratic in the field operator ˆφ. The stochastic theory, on the other hand, is based on the noise kernel 12View Equation, 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 N expansion gives diagrams of order 2 1∕N. This will be illustrated in Section 3.3.1. Furthermore, the retarded propagator also gives information on the commutator
( ) ⟨[ˆhab(x ),ˆhcd(y )]⟩ = 16πiG Greatbcd(y,x ) − Greatbcd(x,y ) , (19 )
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 n-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, 374Jump To The Next Citation Point]. 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, 66Jump To The Next Citation Point].

  Go to previous page Go up Go to next page