Discussion

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 $e −|y|∕LP$ 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 $L2 ∕|y|2 P$ , since the action for the free graviton goes like $∫ 4 −2 Sh ∼ d xL P h□h$ .

For background solutions of semiclassical gravity with other scales present apart from the Planck scales (for instance, for matter fields in a thermal state), stress-energy 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 non-analytic 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 $ℏ$ , the gravitational correlation functions could not be obtained from an expansion in $ℏ$ . 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 $ℏ$ 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.